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Theorem lmhmlem 15786
Description: Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlem.k  |-  K  =  (Scalar `  S )
lmhmlem.l  |-  L  =  (Scalar `  T )
Assertion
Ref Expression
lmhmlem  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K ) ) )

Proof of Theorem lmhmlem
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlem.k . . 3  |-  K  =  (Scalar `  S )
2 lmhmlem.l . . 3  |-  L  =  (Scalar `  T )
3 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
4 eqid 2283 . . 3  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2283 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
6 eqid 2283 . . 3  |-  ( .s
`  T )  =  ( .s `  T
)
71, 2, 3, 4, 5, 6islmhm 15784 . 2  |-  ( F  e.  ( S LMHom  T
)  <->  ( ( S  e.  LMod  /\  T  e. 
LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K  /\  A. a  e.  ( Base `  K
) A. b  e.  ( Base `  S
) ( F `  ( a ( .s
`  S ) b ) )  =  ( a ( .s `  T ) ( F `
 b ) ) ) ) )
8 3simpa 952 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  L  =  K  /\  A. a  e.  ( Base `  K
) A. b  e.  ( Base `  S
) ( F `  ( a ( .s
`  S ) b ) )  =  ( a ( .s `  T ) ( F `
 b ) ) )  ->  ( F  e.  ( S  GrpHom  T )  /\  L  =  K ) )
98anim2i 552 . 2  |-  ( ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K  /\  A. a  e.  ( Base `  K ) A. b  e.  ( Base `  S
) ( F `  ( a ( .s
`  S ) b ) )  =  ( a ( .s `  T ) ( F `
 b ) ) ) )  ->  (
( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K )
) )
107, 9sylbi 187 1  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  lmhmsca  15787  lmghm  15788  lmhmlmod2  15789  lmhmlmod1  15790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-lmhm 15779
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