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Theorem idlmhm 15897
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 2358 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2358 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
4 eqid 2358 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
5 id 19 . 2  |-  ( M  e.  LMod  ->  M  e. 
LMod )
6 eqidd 2359 . 2  |-  ( M  e.  LMod  ->  (Scalar `  M )  =  (Scalar `  M ) )
7 lmodgrp 15733 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Grp )
81idghm 14797 . . 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 15743 . . . . 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 5795 . . . 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 5795 . . . . 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 5961 . . 3  |-  ( ( M  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) ( (  _I  |`  B ) `
 y ) )  =  ( x ( .s `  M ) y ) )
1713, 16eqtr4d 2393 . 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 15895 1  |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M LMHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    _I cid 4386    |` cres 4773   ` cfv 5337  (class class class)co 5945   Basecbs 13245  Scalarcsca 13308   .scvsca 13309   Grpcgrp 14461    GrpHom cghm 14779   LModclmod 15726   LMHom clmhm 15875
This theorem is referenced by:  idnmhm  18365  mendrng  26823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-mnd 14466  df-grp 14588  df-ghm 14780  df-lmod 15728  df-lmhm 15878
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