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Theorem idlmhm 15798
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 2283 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2283 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
4 eqid 2283 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
5 id 19 . 2  |-  ( M  e.  LMod  ->  M  e. 
LMod )
6 eqidd 2284 . 2  |-  ( M  e.  LMod  ->  (Scalar `  M )  =  (Scalar `  M ) )
7 lmodgrp 15634 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Grp )
81idghm 14698 . . 3  |-  ( M  e.  Grp  ->  (  _I  |`  B )  e.  ( M  GrpHom  M ) )
97, 8syl 15 . 2  |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M  GrpHom  M ) )
101, 3, 2, 4lmodvscl 15644 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  B )  ->  ( x ( .s
`  M ) y )  e.  B )
11103expb 1152 . . . 4  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) y )  e.  B )
12 fvresi 5711 . . . 4  |-  ( ( x ( .s `  M ) y )  e.  B  ->  (
(  _I  |`  B ) `
 ( x ( .s `  M ) y ) )  =  ( x ( .s
`  M ) y ) )
1311, 12syl 15 . . 3  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( (  _I  |`  B ) `
 ( x ( .s `  M ) y ) )  =  ( x ( .s
`  M ) y ) )
14 fvresi 5711 . . . . 5  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
1514ad2antll 709 . . . 4  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( (  _I  |`  B ) `
 y )  =  y )
1615oveq2d 5874 . . 3  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) ( (  _I  |`  B ) `
 y ) )  =  ( x ( .s `  M ) y ) )
1713, 16eqtr4d 2318 . 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 15796 1  |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M LMHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    _I cid 4304    |` cres 4691   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   Grpcgrp 14362    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  idnmhm  18263  mendrng  27500
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mnd 14367  df-grp 14489  df-ghm 14681  df-lmod 15629  df-lmhm 15779
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