:: LMOD_5  semantic presentation begin  

















definitionlet "R" be  ($#~v2_struct_0  "non"   ($#v2_struct_0 :::"empty"::: )   )    ($#l6_algstr_0 :::"doubleLoopStr"::: )  ; let "V" be  ($#~v2_struct_0  "non"   ($#v2_struct_0 :::"empty"::: )   )    ($#l1_vectsp_1 :::"VectSpStr"::: )   "over" (Set (Const "R")); let "IT" be   ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "IT" is  :::"linearly-independent":::  means :: LMOD_5:def 1 (Bool  "for" (Set (Var "l")) "being"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" "IT"  "st" (Bool (Bool (Set   ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k4_struct_0 :::"0."::: )  "V"))) "holds"  (Bool (Set   ($#k1_vectsp_6 :::"Carrier"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  ))); end; 






:: deftheorem    defines :::"linearly-independent"::: LMOD_5:def 1 :  (Bool  "for" (Set (Var "R")) "being"  ($#~v2_struct_0  "non"   ($#v2_struct_0 :::"empty"::: )   )    ($#l6_algstr_0 :::"doubleLoopStr"::: )    (Bool  "for" (Set (Var "V")) "being"  ($#~v2_struct_0  "non"   ($#v2_struct_0 :::"empty"::: )   )    ($#l1_vectsp_1 :::"VectSpStr"::: )   "over" (Set (Var "R"))  (Bool  "for" (Set (Var "IT")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "IT")) "is"   ($#v1_lmod_5 :::"linearly-independent"::: )  ) "iff" (Bool  "for" (Set (Var "l")) "being"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "IT"))  "st" (Bool (Bool (Set   ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V"))))) "holds"  (Bool (Set   ($#k1_vectsp_6 :::"Carrier"::: )  (Set (Var "l")))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  )))  ")" )))); 
 notationlet "R" be  ($#~v2_struct_0  "non"   ($#v2_struct_0 :::"empty"::: )   )    ($#l6_algstr_0 :::"doubleLoopStr"::: )  ; let "V" be  ($#~v2_struct_0  "non"   ($#v2_struct_0 :::"empty"::: )   )    ($#l1_vectsp_1 :::"VectSpStr"::: )   "over" (Set (Const "R")); let "IT" be   ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); antonym  :::"linearly-dependent"::: "IT" for  :::"linearly-independent"::: ; end; 
theorem :: LMOD_5:1 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"Ring":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "A"))  ($#r1_tarski :::"c="::: )  (Set (Var "B"))) & (Bool (Set (Var "B")) "is"   ($#v1_lmod_5 :::"linearly-independent"::: )  )) "holds"  (Bool (Set (Var "A")) "is"   ($#v1_lmod_5 :::"linearly-independent"::: )  )))) ; 
 
theorem :: LMOD_5:2 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"Ring":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "R")))  ($#r1_hidden :::"<>"::: )  (Set   ($#k5_struct_0 :::"1."::: )  (Set (Var "R")))) & (Bool (Set (Var "A")) "is"   ($#v1_lmod_5 :::"linearly-independent"::: )  )) "holds"  (Bool  "not" (Bool (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V")))  ($#r2_hidden :::"in"::: )  (Set (Var "A"))))))) ; 
 
theorem :: LMOD_5:3 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"Ring":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R")) "holds"  (Bool (Set   ($#k1_subset_1 :::"{}"::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "V")))) "is"   ($#v1_lmod_5 :::"linearly-independent"::: )  ))) ; 
 
theorem :: LMOD_5:4 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"Ring":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "v1")) "," (Set (Var "v2")) "being"   ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "R")))  ($#r1_hidden :::"<>"::: )  (Set   ($#k5_struct_0 :::"1."::: )  (Set (Var "R")))) & (Bool (Set  ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) ) "is"   ($#v1_lmod_5 :::"linearly-independent"::: )  )) "holds"  (Bool  "("  (Bool (Set (Var "v1"))  ($#r1_hidden :::"<>"::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V")))) & (Bool (Set (Var "v2"))  ($#r1_hidden :::"<>"::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "V"))))  ")" )))) ; 
 
theorem :: LMOD_5:5 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"Ring":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "v")) "being"   ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "R")))  ($#r1_hidden :::"<>"::: )  (Set   ($#k5_struct_0 :::"1."::: )  (Set (Var "R"))))) "holds"  (Bool  "("  (Bool (Set  ($#k7_domain_1 :::"{"::: ) (Set (Var "v")) "," (Set  "("  ($#k4_struct_0 :::"0."::: )  (Set (Var "V")) ")" ) ($#k7_domain_1 :::"}"::: ) ) "is"   ($#v1_lmod_5 :::"linearly-dependent"::: )  ) & (Bool (Set  ($#k7_domain_1 :::"{"::: ) (Set  "("  ($#k4_struct_0 :::"0."::: )  (Set (Var "V")) ")" ) "," (Set (Var "v")) ($#k7_domain_1 :::"}"::: ) ) "is"   ($#v1_lmod_5 :::"linearly-dependent"::: )  )  ")" )))) ; 
 
theorem :: LMOD_5:6 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "L")) "being"    ($#m1_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "V"))  (Bool  "for" (Set (Var "a")) "being"   ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R"))  "st" (Bool (Bool (Set (Var "a"))  ($#r1_hidden :::"<>"::: )  (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "R"))))) "holds"  (Bool (Set   ($#k1_vectsp_6 :::"Carrier"::: )  (Set  "(" (Set (Var "a"))  ($#k6_vectsp_6 :::"*"::: )  (Set (Var "L")) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k1_vectsp_6 :::"Carrier"::: )  (Set (Var "L")))))))) ; 
 
theorem :: LMOD_5:7 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "L")) "being"    ($#m1_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "V"))  (Bool  "for" (Set (Var "a")) "being"   ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) "holds"  (Bool (Set   ($#k4_vectsp_6 :::"Sum"::: )  (Set  "(" (Set (Var "a"))  ($#k6_vectsp_6 :::"*"::: )  (Set (Var "L")) ")" ))  ($#r1_hidden :::"="::: )  (Set (Set (Var "a"))  ($#k4_vectsp_1 :::"*"::: )  (Set  "("  ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "L")) ")" ))))))) ; 
 
















definitionlet "R" be   ($#l6_algstr_0 :::"domRing":::); let "V" be   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Const "R")); let "A" be   ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); func  :::"Lin"::: "A" ->   ($#v7_vectsp_1 :::"strict"::: )     ($#m1_vectsp_4 :::"Subspace"::: )   "of" "V" means :: LMOD_5:def 2 (Bool (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" it)  ($#r1_hidden :::"="::: )   "{"  (Set  "("  ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")) ")" ) where l "is"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" "A" : (Bool verum)  "}"  ); end; 







:: deftheorem    defines :::"Lin"::: LMOD_5:def 2 :  (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  (Bool  "for" (Set (Var "b4")) "being"   ($#v7_vectsp_1 :::"strict"::: )     ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "b4"))  ($#r1_hidden :::"="::: )  (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")))) "iff" (Bool (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "b4")))  ($#r1_hidden :::"="::: )   "{"  (Set  "("  ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")) ")" ) where l "is"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "A")) : (Bool verum)  "}"  )  ")" ))))); 
 
theorem :: LMOD_5:8 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"   (Bool  "("  (Bool (Set (Var "x"))  ($#r1_struct_0 :::"in"::: )  (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")))) "iff" (Bool  "ex" (Set (Var "l")) "being"    ($#m2_vectsp_6 :::"Linear_Combination"::: )   "of" (Set (Var "A")) "st" (Bool (Set (Var "x"))  ($#r1_hidden :::"="::: )  (Set   ($#k4_vectsp_6 :::"Sum"::: )  (Set (Var "l")))))  ")" ))))) ; 
 
theorem :: LMOD_5:9 (Bool  "for" (Set (Var "x")) "being"    ($#m1_hidden :::"set"::: )    (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "x"))  ($#r2_hidden :::"in"::: )  (Set (Var "A")))) "holds"  (Bool (Set (Var "x"))  ($#r1_struct_0 :::"in"::: )  (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")))))))) ; 
 
theorem :: LMOD_5:10 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R")) "holds"  (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set  "("  ($#k1_subset_1 :::"{}"::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "V"))) ")" ))  ($#r1_hidden :::"="::: )  (Set   ($#k1_vectsp_4 :::"(0)."::: )  (Set (Var "V")))))) ; 
 
theorem :: LMOD_5:11 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "holds"  (Bool  "("   "not" (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set   ($#k1_vectsp_4 :::"(0)."::: )  (Set (Var "V")))) "or" (Bool (Set (Var "A"))  ($#r1_hidden :::"="::: )  (Set   ($#k1_xboole_0 :::"{}"::: )  )) "or" (Bool (Set (Var "A"))  ($#r1_hidden :::"="::: )  (Set  ($#k6_domain_1 :::"{"::: ) (Set  "("  ($#k4_struct_0 :::"0."::: )  (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) ))  ")" )))) ; 
 
theorem :: LMOD_5:12 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  (Bool  "for" (Set (Var "W")) "being"   ($#v7_vectsp_1 :::"strict"::: )     ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "R")))  ($#r1_hidden :::"<>"::: )  (Set   ($#k5_struct_0 :::"1."::: )  (Set (Var "R")))) & (Bool (Set (Var "A"))  ($#r1_hidden :::"="::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "W"))))) "holds"  (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set (Var "W"))))))) ; 
 
theorem :: LMOD_5:13 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#v7_vectsp_1 :::"strict"::: )    ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k4_struct_0 :::"0."::: )  (Set (Var "R")))  ($#r1_hidden :::"<>"::: )  (Set   ($#k5_struct_0 :::"1."::: )  (Set (Var "R")))) & (Bool (Set (Var "A"))  ($#r1_hidden :::"="::: )  (Set  "the"  ($#u1_struct_0 :::"carrier"::: )  "of" (Set (Var "V"))))) "holds"  (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set (Var "V")))))) ; 
 
theorem :: LMOD_5:14 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set (Var "A"))  ($#r1_tarski :::"c="::: )  (Set (Var "B")))) "holds"  (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A"))) "is"    ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "B"))))))) ; 
 
theorem :: LMOD_5:15 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#v7_vectsp_1 :::"strict"::: )    ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V"))  "st" (Bool (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")))  ($#r1_hidden :::"="::: )  (Set (Var "V"))) & (Bool (Set (Var "A"))  ($#r1_tarski :::"c="::: )  (Set (Var "B")))) "holds"  (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "B")))  ($#r1_hidden :::"="::: )  (Set (Var "V")))))) ; 
 
theorem :: LMOD_5:16 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"  (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set  "(" (Set (Var "A"))  ($#k4_subset_1 :::"\/"::: )  (Set (Var "B")) ")" ))  ($#r1_hidden :::"="::: )  (Set (Set  "("  ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")) ")" )  ($#k1_vectsp_5 :::"+"::: )  (Set  "("  ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "B")) ")" )))))) ; 
 
theorem :: LMOD_5:17 (Bool  "for" (Set (Var "R")) "being"   ($#l6_algstr_0 :::"domRing":::)  (Bool  "for" (Set (Var "V")) "being"   ($#l1_vectsp_1 :::"LeftMod":::) "of" (Set (Var "R"))  (Bool  "for" (Set (Var "A")) "," (Set (Var "B")) "being"   ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds"  (Bool (Set   ($#k1_lmod_5 :::"Lin"::: )  (Set  "(" (Set (Var "A"))  ($#k9_subset_1 :::"/\"::: )  (Set (Var "B")) ")" )) "is"    ($#m1_vectsp_4 :::"Subspace"::: )   "of" (Set (Set  "("  ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "A")) ")" )  ($#k2_vectsp_5 :::"/\"::: )  (Set  "("  ($#k1_lmod_5 :::"Lin"::: )  (Set (Var "B")) ")" )))))) ;