Theorem issgrp 18807

Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)

Hypotheses

Ref	Expression
issgrp.b	⊢ 𝐵 = (Base‘𝑀)
issgrp.o	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
issgrp	⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥, ⚬ ,𝑦,𝑧

Proof of Theorem issgrp

Dummy variables 𝑏 𝑔 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6903	. . 3 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
2		fveq2 6888	. . . 4 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
3		issgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
4	2, 3	eqtr4di 2819	. . 3 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
5		fvexd 6903	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) ∈ V)
6		fveq2 6888	. . . . . 6 ⊢ (𝑔 = 𝑀 → (+g‘𝑔) = (+g‘𝑀))
7	6	adantr 486	. . . . 5 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) = (+g‘𝑀))
8		issgrp.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
9	7, 8	eqtr4di 2819	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) = ⚬ )
10		simplr 781	. . . . 5 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵)
11		id 23	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑜 = ⚬ )
12		oveq 7429	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
13		eqidd 2767	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑧 = 𝑧)
14	11, 12, 13	oveq123d 7444	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 ⚬ 𝑦) ⚬ 𝑧))
15		eqidd 2767	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑥 = 𝑥)
16		oveq 7429	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑦𝑜𝑧) = (𝑦 ⚬ 𝑧))
17	11, 15, 16	oveq123d 7444	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
18	14, 17	eqeq12d 2782	. . . . . . . 8 ⊢ (𝑜 = ⚬ → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
19	18	adantl 487	. . . . . . 7 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
20	10, 19	raleqbidv 3341	. . . . . 6 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
21	10, 20	raleqbidv 3341	. . . . 5 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
22	10, 21	raleqbidv 3341	. . . 4 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
23	5, 9, 22	sbcied2 3791	. . 3 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → ([(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
24	1, 4, 23	sbcied2 3791	. 2 ⊢ (𝑔 = 𝑀 → ([(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
25		df-sgrp 18806	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
26	24, 25	elrab2 3657	1 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  Vcvv 3458  [wsbc 3747  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  Mgmcmgm 18721  Smgrpcsgrp 18805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-sgrp 18806

This theorem is used by:  issgrpv  18808  issgrpn0  18809  isnsgrp  18810  sgrpmgm  18811  sgrpass  18812  sgrp0  18814  sgrp0b  18815  sgrp1  18816  efmndsgrp  18976  smndex1sgrp  19001  sgrp2nmndlem4  19021  rnglidlmsgrp  21417  copissgrp  48973  nnsgrp  48982  sgrpplusgaopALT  49000  sgrp2sgrp  49033  2zrngasgrp  49051  2zrngmsgrp  49058