Theorem sgrppropd 13454

Description: If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.)

Hypotheses

Ref	Expression
sgrppropd.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
sgrppropd.l	⊢ (𝜑 → 𝐿 ∈ 𝑊)
sgrppropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
sgrppropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
sgrppropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))

Assertion

Ref	Expression
sgrppropd	⊢ (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem sgrppropd

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 528	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ Smgrp)
2		simprl 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
3		sgrppropd.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
4	3	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
5	2, 4	eleqtrd 2308	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐾))
6		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
7	6, 4	eleqtrd 2308	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
8		eqid 2229	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		eqid 2229	. . . . . . 7 ⊢ (+g‘𝐾) = (+g‘𝐾)
10	8, 9	sgrpcl 13450	. . . . . 6 ⊢ ((𝐾 ∈ Smgrp ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
11	1, 5, 7, 10	syl3anc 1271	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
12	11, 4	eleqtrrd 2309	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
13	12	ralrimivva 2612	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ Smgrp) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
14	13	ex 115	. 2 ⊢ (𝜑 → (𝐾 ∈ Smgrp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
15		simplr 528	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ Smgrp)
16		simprl 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
17		sgrppropd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
18	17	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
19	16, 18	eleqtrd 2308	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐿))
20		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
21	20, 18	eleqtrd 2308	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
22		eqid 2229	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
23		eqid 2229	. . . . . . 7 ⊢ (+g‘𝐿) = (+g‘𝐿)
24	22, 23	sgrpcl 13450	. . . . . 6 ⊢ ((𝐿 ∈ Smgrp ∧ 𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))
25	15, 19, 21, 24	syl3anc 1271	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))
26		sgrppropd.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
27	26	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
28	25, 27, 18	3eltr4d 2313	. . . 4 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
29	28	ralrimivva 2612	. . 3 ⊢ ((𝜑 ∧ 𝐿 ∈ Smgrp) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
30	29	ex 115	. 2 ⊢ (𝜑 → (𝐿 ∈ Smgrp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
31		sgrppropd.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
32	8, 9	issgrpv 13445	. . . . . 6 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Smgrp ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
33	31, 32	syl 14	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ Smgrp ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
34	33	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ Smgrp ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
35	26	oveqrspc2v 6034	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
36	35	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
37	36	eleq1d 2298	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ↔ (𝑢(+g‘𝐿)𝑣) ∈ 𝐵))
38		simplll 533	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝜑)
39		simplrl 535	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑢 ∈ 𝐵)
40		simplrr 536	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑣 ∈ 𝐵)
41		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
42		ovrspc2v 6033	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
43	39, 40, 41, 42	syl21anc 1270	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
44		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
45	26	oveqrspc2v 6034	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = ((𝑢(+g‘𝐾)𝑣)(+g‘𝐿)𝑤))
46	38, 43, 44, 45	syl12anc 1269	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = ((𝑢(+g‘𝐾)𝑣)(+g‘𝐿)𝑤))
47	38, 39, 40, 35	syl12anc 1269	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
48	47	oveq1d 6022	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐿)𝑤) = ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤))
49	46, 48	eqtrd 2262	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤))
50		ovrspc2v 6033	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝑣(+g‘𝐾)𝑤) ∈ 𝐵)
51	40, 44, 41, 50	syl21anc 1270	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑣(+g‘𝐾)𝑤) ∈ 𝐵)
52	26	oveqrspc2v 6034	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ (𝑣(+g‘𝐾)𝑤) ∈ 𝐵)) → (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐾)𝑤)))
53	38, 39, 51, 52	syl12anc 1269	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐾)𝑤)))
54	26	oveqrspc2v 6034	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
55	38, 40, 44, 54	syl12anc 1269	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
56	55	oveq2d 6023	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐿)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))
57	53, 56	eqtrd 2262	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))
58	49, 57	eqeq12d 2244	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) ↔ ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))))
59	58	ralbidva 2526	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) ↔ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))))
60	37, 59	anbi12d 473	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
61	60	2ralbidva 2552	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
62	3	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐾))
63	62	eleq2d 2299	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ↔ (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾)))
64	62	raleqdv 2734	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))))
65	63, 64	anbi12d 473	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
66	62, 65	raleqbidv 2744	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
67	62, 66	raleqbidv 2744	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
68	17	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐿))
69	68	eleq2d 2299	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ↔ (𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿)))
70	68	raleqdv 2734	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))))
71	69, 70	anbi12d 473	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ ((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
72	68, 71	raleqbidv 2744	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
73	68, 72	raleqbidv 2744	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
74	61, 67, 73	3bitr3d 218	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
75		sgrppropd.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ 𝑊)
76	22, 23	issgrpv 13445	. . . . . . 7 ⊢ (𝐿 ∈ 𝑊 → (𝐿 ∈ Smgrp ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
77	75, 76	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐿 ∈ Smgrp ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
78	77	bicomd 141	. . . . 5 ⊢ (𝜑 → (∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ 𝐿 ∈ Smgrp))
79	78	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ 𝐿 ∈ Smgrp))
80	34, 74, 79	3bitrd 214	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))
81	80	ex 115	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp)))
82	14, 30, 81	pm5.21ndd 710	1 ⊢ (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ‘cfv 5318  (class class class)co 6007  Basecbs 13040  +_gcplusg 13118  Smgrpcsgrp 13442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6010  df-inn 9119  df-2 9177  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-mgm 13397  df-sgrp 13443

This theorem is referenced by:  rngpropd  13926