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Theorem idlmhm 19808
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 2824 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 eqid 2824 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2824 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5 id 22 . 2 (𝑀 ∈ LMod → 𝑀 ∈ LMod)
6 eqidd 2825 . 2 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
7 lmodgrp 19636 . . 3 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
81idghm 18371 . . 3 (𝑀 ∈ Grp → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
97, 8syl 17 . 2 (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
101, 3, 2, 4lmodvscl 19646 . . . . 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 6924 . . . 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 6924 . . . . 5 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
1514ad2antll 728 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
1615oveq2d 7162 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝐵)) → (𝑥( ·𝑠𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠𝑀)𝑦))
1713, 16eqtr4d 2862 . 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 19806 1 (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115   I cid 5447  cres 5545  cfv 6344  (class class class)co 7146  Basecbs 16481  Scalarcsca 16566   ·𝑠 cvsca 16567  Grpcgrp 18101   GrpHom cghm 18353  LModclmod 19629   LMHom clmhm 19786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-mgm 17850  df-sgrp 17899  df-mnd 17910  df-grp 18104  df-ghm 18354  df-lmod 19631  df-lmhm 19789
This theorem is referenced by:  idnmhm  23358  mendring  39992
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