Theorem idmgmhm 45342

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 550	. 2 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ∧ 𝑀 ∈ Mgm))
3		f1oi 6754	. . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
4		f1of 6716	. . . 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 2738	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
8	6, 7	mgmcl 18329	. . . . . . 7 ⊢ ((𝑀 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
9	8	3expb 1119	. . . . . 6 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
10		fvresi 7045	. . . . . 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 7045	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
13		fvresi 7045	. . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
14	12, 13	oveqan12d 7294	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
15	14	adantl 482	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏))
16	11, 15	eqtr4d 2781	. . . 4 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))
17	16	ralrimivva 3123	. . 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 512	. 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 45337	. 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 583	1 ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   I cid 5488   ↾ cres 5591  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Mgmcmgm 18324   MgmHom cmgmhm 45331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-mgm 18326  df-mgmhm 45333

This theorem is referenced by:  idrnghm  45466