Theorem idlmhm 19041

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 2622	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2622	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
4		eqid 2622	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5		id 22	. 2 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ LMod)
6		eqidd 2623	. 2 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
7		lmodgrp 18870	. . 3 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
8	1	idghm 17675	. . 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 18880	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
11	10	3expb 1266	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
12		fvresi 6439	. . . 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 6439	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
15	14	ad2antll 765	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
16	15	oveq2d 6666	. . 3 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑀)(( I ↾ 𝐵)‘𝑦)) = (𝑥( ·𝑠 ‘𝑀)𝑦))
17	13, 16	eqtr4d 2659	. 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 19039	1 ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   I cid 5023   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  Grpcgrp 17422   GrpHom cghm 17657  LModclmod 18863   LMHom clmhm 19019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  idnmhm  22558  mendring  37762