Theorem issgrp 18164

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 6732	. . 3 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
2		fveq2 6717	. . . 4 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
3		issgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
4	2, 3	eqtr4di 2796	. . 3 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
5		fvexd 6732	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) ∈ V)
6		fveq2 6717	. . . . . 6 ⊢ (𝑔 = 𝑀 → (+g‘𝑔) = (+g‘𝑀))
7	6	adantr 484	. . . . 5 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) = (+g‘𝑀))
8		issgrp.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
9	7, 8	eqtr4di 2796	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) = ⚬ )
10		simplr 769	. . . . 5 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵)
11		id 22	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑜 = ⚬ )
12		oveq 7219	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
13		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑧 = 𝑧)
14	11, 12, 13	oveq123d 7234	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 ⚬ 𝑦) ⚬ 𝑧))
15		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑥 = 𝑥)
16		oveq 7219	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑦𝑜𝑧) = (𝑦 ⚬ 𝑧))
17	11, 15, 16	oveq123d 7234	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
18	14, 17	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑜 = ⚬ → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
19	18	adantl 485	. . . . . . 7 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
20	10, 19	raleqbidv 3313	. . . . . 6 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
21	10, 20	raleqbidv 3313	. . . . 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 3313	. . . 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 3741	. . 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 3741	. 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 18163	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
26	24, 25	elrab2 3605	1 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408  [wsbc 3694  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  +_gcplusg 16802  Mgmcmgm 18112  Smgrpcsgrp 18162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-sgrp 18163

This theorem is referenced by:  issgrpv  18165  issgrpn0  18166  isnsgrp  18167  sgrpmgm  18168  sgrpass  18169  sgrp0  18170  sgrp0b  18171  sgrp1  18172  efmndsgrp  18313  smndex1sgrp  18335  sgrp2nmndlem4  18355  copissgrp  45035  nnsgrp  45044  sgrpplusgaopALT  45062  sgrp2sgrp  45095  lidlmsgrp  45157  2zrngasgrp  45171  2zrngmsgrp  45178