Theorem idmhm 18808

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 22	. 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mnd)
2		f1oi 6886	. . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
3		f1of 6848	. . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵)
4	2, 3	mp1i 13	. . 3 ⊢ (𝑀 ∈ Mnd → ( I ↾ 𝐵):𝐵⟶𝐵)
5		idmhm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
6		eqid 2737	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
7	5, 6	mndcl 18755	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
8	7	3expb 1121	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
9		fvresi 7193	. . . . . 6 ⊢ ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏))
10	8, 9	syl 17	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏))
11		fvresi 7193	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
12		fvresi 7193	. . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
13	11, 12	oveqan12d 7450	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
14	13	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
15	10, 14	eqtr4d 2780	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
16	15	ralrimivva 3202	. . 3 ⊢ (𝑀 ∈ Mnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
17		eqid 2737	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
18	5, 17	mndidcl 18762	. . . 4 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
19		fvresi 7193	. . . 4 ⊢ ((0g‘𝑀) ∈ 𝐵 → (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀))
20	18, 19	syl 17	. . 3 ⊢ (𝑀 ∈ Mnd → (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀))
21	4, 16, 20	3jca 1129	. 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 18798	. 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 1345	1 ⊢ (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   I cid 5577   ↾ cres 5687  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  0_gc0g 17484  Mndcmnd 18747   MndHom cmhm 18794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796

This theorem is referenced by:  idrhm  20490