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Theorem idlmhm 20938
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 2725 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 eqid 2725 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2725 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 id 22 . 2 (𝑀 ∈ LMod → 𝑀 ∈ LMod)
6 eqidd 2726 . 2 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
7 lmodgrp 20762 . . 3 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
81idghm 19194 . . 3 (𝑀 ∈ Grp → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
97, 8syl 17 . 2 (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
101, 3, 2, 4lmodvscl 20773 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝐵)
11103expb 1117 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝐵)
12 fvresi 7182 . . . 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 7182 . . . . 5 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
1514ad2antll 727 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
1615oveq2d 7435 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (𝑥( ·𝑠𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠𝑀)𝑦))
1713, 16eqtr4d 2768 . 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 20936 1 (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098   I cid 5575  cres 5680  cfv 6549  (class class class)co 7419  Basecbs 17183  Scalarcsca 17239   ·𝑠 cvsca 17240  Grpcgrp 18898   GrpHom cghm 19175  LModclmod 20755   LMHom clmhm 20916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-ghm 19176  df-lmod 20757  df-lmhm 20919
This theorem is referenced by:  idnmhm  24715  mendring  42755
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