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Theorem lmhmlem 15802
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 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
4 eqid 2296 . . 3  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2296 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
6 eqid 2296 . . 3  |-  ( .s
`  T )  =  ( .s `  T
)
71, 2, 3, 4, 5, 6islmhm 15800 . 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228    GrpHom cghm 14696   LModclmod 15643   LMHom clmhm 15792
This theorem is referenced by:  lmhmsca  15803  lmghm  15804  lmhmlmod2  15805  lmhmlmod1  15806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-lmhm 15795
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