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Theorem idlmhm 16122
Description: The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Hypothesis
Ref Expression
idlmhm.b  |-  B  =  ( Base `  M
)
Assertion
Ref Expression
idlmhm  |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M LMHom  M ) )

Proof of Theorem idlmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idlmhm.b . 2  |-  B  =  ( Base `  M
)
2 eqid 2438 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2438 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
4 eqid 2438 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
5 id 21 . 2  |-  ( M  e.  LMod  ->  M  e. 
LMod )
6 eqidd 2439 . 2  |-  ( M  e.  LMod  ->  (Scalar `  M )  =  (Scalar `  M ) )
7 lmodgrp 15962 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Grp )
81idghm 15026 . . 3  |-  ( M  e.  Grp  ->  (  _I  |`  B )  e.  ( M  GrpHom  M ) )
97, 8syl 16 . 2  |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M  GrpHom  M ) )
101, 3, 2, 4lmodvscl 15972 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  B )  ->  ( x ( .s
`  M ) y )  e.  B )
11103expb 1155 . . . 4  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) y )  e.  B )
12 fvresi 5927 . . . 4  |-  ( ( x ( .s `  M ) y )  e.  B  ->  (
(  _I  |`  B ) `
 ( x ( .s `  M ) y ) )  =  ( x ( .s
`  M ) y ) )
1311, 12syl 16 . . 3  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( (  _I  |`  B ) `
 ( x ( .s `  M ) y ) )  =  ( x ( .s
`  M ) y ) )
14 fvresi 5927 . . . . 5  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
1514ad2antll 711 . . . 4  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( (  _I  |`  B ) `
 y )  =  y )
1615oveq2d 6100 . . 3  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) ( (  _I  |`  B ) `
 y ) )  =  ( x ( .s `  M ) y ) )
1713, 16eqtr4d 2473 . 2  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( (  _I  |`  B ) `
 ( x ( .s `  M ) y ) )  =  ( x ( .s
`  M ) ( (  _I  |`  B ) `
 y ) ) )
181, 2, 2, 3, 3, 4, 5, 5, 6, 9, 17islmhmd 16120 1  |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M LMHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    _I cid 4496    |` cres 4883   ` cfv 5457  (class class class)co 6084   Basecbs 13474  Scalarcsca 13537   .scvsca 13538   Grpcgrp 14690    GrpHom cghm 15008   LModclmod 15955   LMHom clmhm 16100
This theorem is referenced by:  idnmhm  18793  mendrng  27491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-mnd 14695  df-grp 14817  df-ghm 15009  df-lmod 15957  df-lmhm 16103
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