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Theorem idlmhm 16076
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 2408 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2408 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
4 eqid 2408 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
5 id 20 . 2  |-  ( M  e.  LMod  ->  M  e. 
LMod )
6 eqidd 2409 . 2  |-  ( M  e.  LMod  ->  (Scalar `  M )  =  (Scalar `  M ) )
7 lmodgrp 15916 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Grp )
81idghm 14980 . . 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 15926 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  B )  ->  ( x ( .s
`  M ) y )  e.  B )
11103expb 1154 . . . 4  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) y )  e.  B )
12 fvresi 5887 . . . 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 5887 . . . . 5  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
1514ad2antll 710 . . . 4  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( (  _I  |`  B ) `
 y )  =  y )
1615oveq2d 6060 . . 3  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) ( (  _I  |`  B ) `
 y ) )  =  ( x ( .s `  M ) y ) )
1713, 16eqtr4d 2443 . 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 16074 1  |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M LMHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    _I cid 4457    |` cres 4843   ` cfv 5417  (class class class)co 6044   Basecbs 13428  Scalarcsca 13491   .scvsca 13492   Grpcgrp 14644    GrpHom cghm 14962   LModclmod 15909   LMHom clmhm 16054
This theorem is referenced by:  idnmhm  18745  mendrng  27372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-mnd 14649  df-grp 14771  df-ghm 14963  df-lmod 15911  df-lmhm 16057
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