Theorem issgrpd 18617

Description: Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)

Hypotheses

Ref	Expression
issgrpd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
issgrpd.p	⊢ (𝜑 → + = (+g‘𝐺))
issgrpd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
issgrpd.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
issgrpd.v	⊢ (𝜑 → 𝐺 ∈ 𝑉)

Assertion

Ref	Expression
issgrpd	⊢ (𝜑 → 𝐺 ∈ Smgrp)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐺,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   + (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem issgrpd

Step	Hyp	Ref	Expression
1		issgrpd.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
2	1	3expib 1122	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵))
3		issgrpd.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
4	3	eleq2d 2819	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺)))
5	3	eleq2d 2819	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐺)))
6	4, 5	anbi12d 631	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))))
7		issgrpd.p	. . . . . . . 8 ⊢ (𝜑 → + = (+g‘𝐺))
8	7	oveqd 7422	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
9	8, 3	eleq12d 2827	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
10	2, 6, 9	3imtr3d 292	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
11	10	imp 407	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
12		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵))
13		issgrpd.a	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
14	12, 13	sylan2br 595	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
15	14	ex 413	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
16	3	eleq2d 2819	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (Base‘𝐺)))
17	6, 16	anbi12d 631	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑧 ∈ (Base‘𝐺))))
18		eqidd 2733	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 = 𝑧)
19	7, 8, 18	oveq123d 7426	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧))
20		eqidd 2733	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 = 𝑥)
21	7	oveqd 7422	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑧) = (𝑦(+g‘𝐺)𝑧))
22	7, 20, 21	oveq123d 7426	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
23	19, 22	eqeq12d 2748	. . . . . . 7 ⊢ (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
24	15, 17, 23	3imtr3d 292	. . . . . 6 ⊢ (𝜑 → (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
25	24	impl 456	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
26	25	ralrimiva 3146	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
27	11, 26	jca 512	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
28	27	ralrimivva 3200	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
29		issgrpd.v	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
30		eqid 2732	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
31		eqid 2732	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
32	30, 31	issgrpv 18608	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))))
33	29, 32	syl 17	. 2 ⊢ (𝜑 → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))))
34	28, 33	mpbird 256	1 ⊢ (𝜑 → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  +_gcplusg 17193  Smgrpcsgrp 18605

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-mgm 18557  df-sgrp 18606

This theorem is referenced by:  prdssgrpd  18620  isrngd  46658