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Theorem idlmhm 21040
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 ∈ LMod → ( I ↾ 𝐵) ∈ (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 2737 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 eqid 2737 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2737 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 id 22 . 2 (𝑀 ∈ LMod → 𝑀 ∈ LMod)
6 eqidd 2738 . 2 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
7 lmodgrp 20865 . . 3 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
81idghm 19249 . . 3 (𝑀 ∈ Grp → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
97, 8syl 17 . 2 (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
101, 3, 2, 4lmodvscl 20876 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝐵)
11103expb 1121 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝐵)
12 fvresi 7193 . . . 4 ((𝑥( ·𝑠𝑀)𝑦) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑀)𝑦))
1311, 12syl 17 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (( I ↾ 𝐵)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑀)𝑦))
14 fvresi 7193 . . . . 5 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
1514ad2antll 729 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
1615oveq2d 7447 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (𝑥( ·𝑠𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠𝑀)𝑦))
1713, 16eqtr4d 2780 . 2 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (( I ↾ 𝐵)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑀)(( I ↾ 𝐵)‘𝑦)))
181, 2, 2, 3, 3, 4, 5, 5, 6, 9, 17islmhmd 21038 1 (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108   I cid 5577  cres 5687  cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  Grpcgrp 18951   GrpHom cghm 19230  LModclmod 20858   LMHom clmhm 21018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ghm 19231  df-lmod 20860  df-lmhm 21021
This theorem is referenced by:  idnmhm  24775  mendring  43200
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