Theorem idmhm 18722

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 6838	. . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
3		f1of 6800	. . . 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 2729	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
7	5, 6	mndcl 18669	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
8	7	3expb 1120	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
9		fvresi 7147	. . . . . 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 7147	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
12		fvresi 7147	. . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
13	11, 12	oveqan12d 7406	. . . . . 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 2767	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
16	15	ralrimivva 3180	. . 3 ⊢ (𝑀 ∈ Mnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
17		eqid 2729	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
18	5, 17	mndidcl 18676	. . . 4 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
19		fvresi 7147	. . . 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 1128	. 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 18712	. 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   I cid 5532   ↾ cres 5640  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Mndcmnd 18661   MndHom cmhm 18708

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710

This theorem is referenced by:  idrhm  20399