Theorem idmhm 18848

Description: The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)

Hypothesis

Ref	Expression
idmhm.b	⊢ 𝐵 = (Base‘𝑀)

Assertion

Ref	Expression
idmhm	⊢ (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀))

Proof of Theorem idmhm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mnd)
2		f1oi 6859	. . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
3		f1of 6820	. . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵)
4	2, 3	mp1i 14	. . 3 ⊢ (𝑀 ∈ Mnd → ( I ↾ 𝐵):𝐵⟶𝐵)
5		idmhm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
6		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
7	5, 6	mndcl 18795	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
8	7	3expb 1138	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
9		fvresi 7171	. . . . . 6 ⊢ ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏))
10	8, 9	syl 18	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏))
11		fvresi 7171	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
12		fvresi 7171	. . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
13	11, 12	oveqan12d 7429	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
14	13	adantl 486	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
15	10, 14	eqtr4d 2801	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
16	15	ralrimivva 3208	. . 3 ⊢ (𝑀 ∈ Mnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
17		eqid 2763	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
18	5, 17	mndidcl 18802	. . . 4 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
19		fvresi 7171	. . . 4 ⊢ ((0g‘𝑀) ∈ 𝐵 → (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀))
20	18, 19	syl 18	. . 3 ⊢ (𝑀 ∈ Mnd → (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀))
21	4, 16, 20	3jca 1146	. 2 ⊢ (𝑀 ∈ Mnd → (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) ∧ (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀)))
22	5, 5, 6, 6, 17, 17	ismhm 18838	. 2 ⊢ (( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀) ↔ ((𝑀 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) ∧ (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀))))
23	1, 1, 21, 22	syl21anbrc 1363	1 ⊢ (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   I cid 5555   ↾ cres 5663  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  +_gcplusg 17305  0_gc0g 17487  Mndcmnd 18787   MndHom cmhm 18834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836

This theorem is referenced by:  idrhm  20573