Theorem idmgmhm 44408

Description: The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)

Hypothesis

Ref	Expression
idmgmhm.b	⊢ 𝐵 = (Base‘𝑀)

Assertion

Ref	Expression
idmgmhm	⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀))

Proof of Theorem idmgmhm

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

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ Mgm)
2	1	ancri 553	. 2 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ∧ 𝑀 ∈ Mgm))
3		f1oi 6627	. . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
4		f1of 6590	. . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵)
5	3, 4	mp1i 13	. . 3 ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵):𝐵⟶𝐵)
6		idmgmhm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
7		eqid 2798	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
8	6, 7	mgmcl 17847	. . . . . . 7 ⊢ ((𝑀 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
9	8	3expb 1117	. . . . . 6 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
10		fvresi 6912	. . . . . 6 ⊢ ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏))
11	9, 10	syl 17	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏))
12		fvresi 6912	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
13		fvresi 6912	. . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
14	12, 13	oveqan12d 7154	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
15	14	adantl 485	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
16	11, 15	eqtr4d 2836	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
17	16	ralrimivva 3156	. . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
18	5, 17	jca 515	. 2 ⊢ (𝑀 ∈ Mgm → (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏))))
19	6, 6, 7, 7	ismgmhm 44403	. 2 ⊢ (( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀) ↔ ((𝑀 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))))
20	2, 18, 19	sylanbrc 586	1 ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   I cid 5424   ↾ cres 5521  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  Mgmcmgm 17842   MgmHom cmgmhm 44397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-mgm 17844  df-mgmhm 44399

This theorem is referenced by:  idrnghm  44532