Theorem idlmhm 21088

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.

Step	Hyp	Ref	Expression
1		idlmhm.b	. 2 ⊢ 𝐵 = (Base‘𝑀)
2		eqid 2761	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2761	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
4		eqid 2761	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5		id 22	. 2 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ LMod)
6		eqidd 2762	. 2 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
7		lmodgrp 20914	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
8	1	idghm 19254	. . 3 ⊢ (𝑀 ∈ Grp → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
9	7, 8	syl 17	. 2 ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 GrpHom 𝑀))
10	1, 3, 2, 4	lmodvscl 20925	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
11	10	3expb 1132	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
12		fvresi 7153	. . . 4 ⊢ ((𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦))
13	11, 12	syl 17	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦))
14		fvresi 7153	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
15	14	ad2antll 739	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
16	15	oveq2d 7408	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦))
17	13, 16	eqtr4d 2799	. 2 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑀)(( I ↾ 𝐵)‘𝑦)))
18	1, 2, 2, 3, 3, 4, 5, 5, 6, 9, 17	islmhmd 21086	1 ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   I cid 5539   ↾ cres 5647  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  Scalarcsca 17272   ·_𝑠 cvsca 17273  Grpcgrp 18958   GrpHom cghm 19236  LModclmod 20907   LMHom clmhm 21066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-map 8805  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-ghm 19237  df-lmod 20909  df-lmhm 21069

This theorem is referenced by:  idnmhm  24794  mendring  43729