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Theorem islindf5 18112
Description: A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Hypotheses
Ref Expression
islindf5.f  |-  F  =  ( R freeLMod  I )
islindf5.b  |-  B  =  ( Base `  F
)
islindf5.c  |-  C  =  ( Base `  T
)
islindf5.v  |-  .x.  =  ( .s `  T )
islindf5.e  |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )
islindf5.t  |-  ( ph  ->  T  e.  LMod )
islindf5.i  |-  ( ph  ->  I  e.  X )
islindf5.r  |-  ( ph  ->  R  =  (Scalar `  T ) )
islindf5.a  |-  ( ph  ->  A : I --> C )
Assertion
Ref Expression
islindf5  |-  ( ph  ->  ( A LIndF  T  <->  E : B -1-1-> C ) )
Distinct variable groups:    ph, x    x, A    x, B    x, C    x, F    x, I    x, R    x, T    x,  .x.    x, X
Allowed substitution hint:    E( x)

Proof of Theorem islindf5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 islindf5.t . . . 4  |-  ( ph  ->  T  e.  LMod )
2 islindf5.i . . . 4  |-  ( ph  ->  I  e.  X )
3 islindf5.a . . . 4  |-  ( ph  ->  A : I --> C )
4 islindf5.c . . . . 5  |-  C  =  ( Base `  T
)
5 eqid 2435 . . . . 5  |-  (Scalar `  T )  =  (Scalar `  T )
6 islindf5.v . . . . 5  |-  .x.  =  ( .s `  T )
7 eqid 2435 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
8 eqid 2435 . . . . 5  |-  ( 0g
`  (Scalar `  T )
)  =  ( 0g
`  (Scalar `  T )
)
9 eqid 2435 . . . . 5  |-  ( Base `  ( (Scalar `  T
) freeLMod  I ) )  =  ( Base `  (
(Scalar `  T ) freeLMod  I ) )
104, 5, 6, 7, 8, 9islindf4 18111 . . . 4  |-  ( ( T  e.  LMod  /\  I  e.  X  /\  A :
I --> C )  -> 
( A LIndF  T  <->  A. y  e.  ( Base `  (
(Scalar `  T ) freeLMod  I ) ) ( ( T  gsumg  ( y  oF  .x.  A ) )  =  ( 0g `  T )  ->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) ) ) )
111, 2, 3, 10syl3anc 1213 . . 3  |-  ( ph  ->  ( A LIndF  T  <->  A. y  e.  ( Base `  (
(Scalar `  T ) freeLMod  I ) ) ( ( T  gsumg  ( y  oF  .x.  A ) )  =  ( 0g `  T )  ->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) ) ) )
12 oveq1 6089 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  oF  .x.  A )  =  ( y  oF  .x.  A ) )
1312oveq2d 6098 . . . . . . . . 9  |-  ( x  =  y  ->  ( T  gsumg  ( x  oF  .x.  A ) )  =  ( T  gsumg  ( y  oF  .x.  A
) ) )
14 islindf5.e . . . . . . . . 9  |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  oF  .x.  A ) ) )
15 ovex 6107 . . . . . . . . 9  |-  ( T 
gsumg  ( y  oF  .x.  A ) )  e.  _V
1613, 14, 15fvmpt 5764 . . . . . . . 8  |-  ( y  e.  B  ->  ( E `  y )  =  ( T  gsumg  ( y  oF  .x.  A
) ) )
1716adantl 463 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  ( E `  y )  =  ( T  gsumg  ( y  oF  .x.  A
) ) )
1817eqeq1d 2443 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( E `  y
)  =  ( 0g
`  T )  <->  ( T  gsumg  ( y  oF  .x.  A ) )  =  ( 0g `  T
) ) )
19 islindf5.r . . . . . . . . . . 11  |-  ( ph  ->  R  =  (Scalar `  T ) )
205lmodrng 16882 . . . . . . . . . . . 12  |-  ( T  e.  LMod  ->  (Scalar `  T )  e.  Ring )
211, 20syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  T )  e.  Ring )
2219, 21eqeltrd 2509 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
23 islindf5.f . . . . . . . . . . 11  |-  F  =  ( R freeLMod  I )
24 eqid 2435 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
2523, 24frlm0 18023 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  X )  ->  (
I  X.  { ( 0g `  R ) } )  =  ( 0g `  F ) )
2622, 2, 25syl2anc 656 . . . . . . . . 9  |-  ( ph  ->  ( I  X.  {
( 0g `  R
) } )  =  ( 0g `  F
) )
2719fveq2d 5685 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  (Scalar `  T )
) )
2827sneqd 3879 . . . . . . . . . 10  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  (Scalar `  T ) ) } )
2928xpeq2d 4853 . . . . . . . . 9  |-  ( ph  ->  ( I  X.  {
( 0g `  R
) } )  =  ( I  X.  {
( 0g `  (Scalar `  T ) ) } ) )
3026, 29eqtr3d 2469 . . . . . . . 8  |-  ( ph  ->  ( 0g `  F
)  =  ( I  X.  { ( 0g
`  (Scalar `  T )
) } ) )
3130adantr 462 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  ( 0g `  F )  =  ( I  X.  {
( 0g `  (Scalar `  T ) ) } ) )
3231eqeq2d 2446 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
y  =  ( 0g
`  F )  <->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) ) )
3318, 32imbi12d 320 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( ( E `  y )  =  ( 0g `  T )  ->  y  =  ( 0g `  F ) )  <->  ( ( T 
gsumg  ( y  oF  .x.  A ) )  =  ( 0g `  T )  ->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) ) ) )
3433ralbidva 2723 . . . 4  |-  ( ph  ->  ( A. y  e.  B  ( ( E `
 y )  =  ( 0g `  T
)  ->  y  =  ( 0g `  F ) )  <->  A. y  e.  B  ( ( T  gsumg  ( y  oF  .x.  A
) )  =  ( 0g `  T )  ->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) ) ) )
3519eqcomd 2440 . . . . . . . . 9  |-  ( ph  ->  (Scalar `  T )  =  R )
3635oveq1d 6097 . . . . . . . 8  |-  ( ph  ->  ( (Scalar `  T
) freeLMod  I )  =  ( R freeLMod  I ) )
3736, 23syl6eqr 2485 . . . . . . 7  |-  ( ph  ->  ( (Scalar `  T
) freeLMod  I )  =  F )
3837fveq2d 5685 . . . . . 6  |-  ( ph  ->  ( Base `  (
(Scalar `  T ) freeLMod  I ) )  =  (
Base `  F )
)
39 islindf5.b . . . . . 6  |-  B  =  ( Base `  F
)
4038, 39syl6eqr 2485 . . . . 5  |-  ( ph  ->  ( Base `  (
(Scalar `  T ) freeLMod  I ) )  =  B )
4140raleqdv 2915 . . . 4  |-  ( ph  ->  ( A. y  e.  ( Base `  (
(Scalar `  T ) freeLMod  I ) ) ( ( T  gsumg  ( y  oF  .x.  A ) )  =  ( 0g `  T )  ->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) )  <->  A. y  e.  B  ( ( T  gsumg  ( y  oF  .x.  A ) )  =  ( 0g `  T )  ->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) ) ) )
4234, 41bitr4d 256 . . 3  |-  ( ph  ->  ( A. y  e.  B  ( ( E `
 y )  =  ( 0g `  T
)  ->  y  =  ( 0g `  F ) )  <->  A. y  e.  (
Base `  ( (Scalar `  T ) freeLMod  I ) ) ( ( T 
gsumg  ( y  oF  .x.  A ) )  =  ( 0g `  T )  ->  y  =  ( I  X.  { ( 0g `  (Scalar `  T ) ) } ) ) ) )
4311, 42bitr4d 256 . 2  |-  ( ph  ->  ( A LIndF  T  <->  A. y  e.  B  ( ( E `  y )  =  ( 0g `  T )  ->  y  =  ( 0g `  F ) ) ) )
4423, 39, 4, 6, 14, 1, 2, 19, 3frlmup1 18070 . . 3  |-  ( ph  ->  E  e.  ( F LMHom 
T ) )
45 lmghm 17036 . . 3  |-  ( E  e.  ( F LMHom  T
)  ->  E  e.  ( F  GrpHom  T ) )
46 eqid 2435 . . . 4  |-  ( 0g
`  F )  =  ( 0g `  F
)
4739, 4, 46, 7ghmf1 15757 . . 3  |-  ( E  e.  ( F  GrpHom  T )  ->  ( E : B -1-1-> C  <->  A. y  e.  B  ( ( E `  y )  =  ( 0g `  T )  ->  y  =  ( 0g `  F ) ) ) )
4844, 45, 473syl 20 . 2  |-  ( ph  ->  ( E : B -1-1-> C  <->  A. y  e.  B  ( ( E `  y )  =  ( 0g `  T )  ->  y  =  ( 0g `  F ) ) ) )
4943, 48bitr4d 256 1  |-  ( ph  ->  ( A LIndF  T  <->  E : B -1-1-> C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1757   A.wral 2707   {csn 3867   class class class wbr 4282    e. cmpt 4340    X. cxp 4827   -->wf 5404   -1-1->wf1 5405   ` cfv 5408  (class class class)co 6082    oFcof 6309   Basecbs 14159  Scalarcsca 14226   .scvsca 14227   0gc0g 14363    gsumg cgsu 14364    GrpHom cghm 15726   Ringcrg 16579   LModclmod 16874   LMHom clmhm 17024   freeLMod cfrlm 18015   LIndF clindf 18077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-inf2 7837  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-iin 4164  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-se 4669  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-isom 5417  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6311  df-om 6468  df-1st 6568  df-2nd 6569  df-supp 6682  df-recs 6820  df-rdg 6854  df-1o 6910  df-oadd 6914  df-er 7091  df-map 7206  df-ixp 7254  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-fsupp 7611  df-sup 7681  df-oi 7714  df-card 8099  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-nn 10313  df-2 10370  df-3 10371  df-4 10372  df-5 10373  df-6 10374  df-7 10375  df-8 10376  df-9 10377  df-10 10378  df-n0 10570  df-z 10637  df-dec 10746  df-uz 10852  df-fz 11427  df-fzo 11535  df-seq 11793  df-hash 12090  df-struct 14161  df-ndx 14162  df-slot 14163  df-base 14164  df-sets 14165  df-ress 14166  df-plusg 14236  df-mulr 14237  df-sca 14239  df-vsca 14240  df-ip 14241  df-tset 14242  df-ple 14243  df-ds 14245  df-hom 14247  df-cco 14248  df-0g 14365  df-gsum 14366  df-prds 14371  df-pws 14373  df-mre 14509  df-mrc 14510  df-acs 14512  df-mnd 15400  df-mhm 15449  df-submnd 15450  df-grp 15527  df-minusg 15528  df-sbg 15529  df-mulg 15530  df-subg 15660  df-ghm 15727  df-cntz 15817  df-cmn 16261  df-abl 16262  df-mgp 16568  df-rng 16582  df-ur 16584  df-subrg 16789  df-lmod 16876  df-lss 16938  df-lsp 16977  df-lmhm 17027  df-lbs 17080  df-sra 17177  df-rgmod 17178  df-nzr 17264  df-dsmm 18001  df-frlm 18016  df-uvc 18052  df-lindf 18079
This theorem is referenced by:  indlcim  18113
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