Theorem sgrppropd 18618

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 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ Smgrp)
2		simprl 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
3		sgrppropd.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
4	3	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
5	2, 4	eleqtrd 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐾))
6		simprr 771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
7	6, 4	eleqtrd 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
8		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		eqid 2732	. . . . . . 7 ⊢ (+g‘𝐾) = (+g‘𝐾)
10	8, 9	sgrpcl 18613	. . . . . 6 ⊢ ((𝐾 ∈ Smgrp ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
11	1, 5, 7, 10	syl3anc 1371	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
12	11, 4	eleqtrrd 2836	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
13	12	ralrimivva 3200	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ Smgrp) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
14	13	ex 413	. 2 ⊢ (𝜑 → (𝐾 ∈ Smgrp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
15		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ Smgrp)
16		simprl 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
17		sgrppropd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
18	17	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
19	16, 18	eleqtrd 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐿))
20		simprr 771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
21	20, 18	eleqtrd 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
22		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
23		eqid 2732	. . . . . . 7 ⊢ (+g‘𝐿) = (+g‘𝐿)
24	22, 23	sgrpcl 18613	. . . . . 6 ⊢ ((𝐿 ∈ Smgrp ∧ 𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))
25	15, 19, 21, 24	syl3anc 1371	. . . . 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 713	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
28	25, 27, 18	3eltr4d 2848	. . . 4 ⊢ (((𝜑 ∧ 𝐿 ∈ Smgrp) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
29	28	ralrimivva 3200	. . 3 ⊢ ((𝜑 ∧ 𝐿 ∈ Smgrp) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
30	29	ex 413	. 2 ⊢ (𝜑 → (𝐿 ∈ Smgrp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
31		sgrppropd.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
32	8, 9	issgrpv 18608	. . . . . 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 17	. . . . 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 481	. . . 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 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
36	35	adantlr 713	. . . . . . . 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 2818	. . . . . . 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 773	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝜑)
39		simplrl 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑢 ∈ 𝐵)
40		simplrr 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑣 ∈ 𝐵)
41		simpllr 774	. . . . . . . . . . . 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 7431	. . . . . . . . . . . 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 836	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
44		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
45	26	oveqrspc2v 7432	. . . . . . . . . . 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 835	. . . . . . . . . 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 835	. . . . . . . . . . 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 7420	. . . . . . . . . 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 2772	. . . . . . . . 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 7431	. . . . . . . . . . . 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 836	. . . . . . . . . . 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 7432	. . . . . . . . . . 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 835	. . . . . . . . . 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 7432	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
55	38, 40, 44, 54	syl12anc 835	. . . . . . . . . . 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 7421	. . . . . . . . . 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 2772	. . . . . . . . 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 2748	. . . . . . . 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 3175	. . . . . . 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 631	. . . . . 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 3216	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐾))
63	62	eleq2d 2819	. . . . . . . 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 3325	. . . . . . . 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 631	. . . . . . 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 3342	. . . . . 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 3342	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐿))
69	68	eleq2d 2819	. . . . . . . 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 3325	. . . . . . . 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 631	. . . . . . 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 3342	. . . . . 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 3342	. . . . 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 308	. . . 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 18608	. . . . . . 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 17	. . . . . 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 222	. . . . 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 481	. . . 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 304	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))
81	80	ex 413	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp)))
82	14, 30, 81	pm5.21ndd 380	1 ⊢ (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = 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-rex 3071  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:  rngpropd  46659  prdsrngd  46663