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Theorem rngpropd 20146

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

**Hypotheses**
Ref	Expression
rngpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
rngpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
rngpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
rngpropd.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝐿)𝑦))

**Assertion**
Ref	Expression
rngpropd	⊢ (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))

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

Proof of Theorem rngpropd Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝜑)
2		simprll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
3		simplrl 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐾 ∈ Abel)
4		simprlr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
5		rngpropd.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
6	5	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
7	4, 6	eleqtrd 2839	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ (Base‘𝐾))
8		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
9	8, 6	eleqtrd 2839	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		ablgrp 19751	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Abel → 𝐾 ∈ Grp)
11		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
13	11, 12	grpcl 18908	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
14	10, 13	syl3an1 1164	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
15	3, 7, 9, 14	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
16	15, 6	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) ∈ 𝐵)
17		rngpropd.4	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝐿)𝑦))
18	17	oveqrspc2v 7387	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ (𝑣(+_g‘𝐾)𝑤) ∈ 𝐵)) → (𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = (𝑢(._r‘𝐿)(𝑣(+_g‘𝐾)𝑤)))
19	1, 2, 16, 18	syl12anc 837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = (𝑢(._r‘𝐿)(𝑣(+_g‘𝐾)𝑤)))
20		rngpropd.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
21	20	oveqrspc2v 7387	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
22	1, 4, 8, 21	syl12anc 837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
23	22	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐿)(𝑣(+_g‘𝐾)𝑤)) = (𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)))
24	19, 23	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = (𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)))
25		simplrr 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Smgrp)
26	2, 6	eleqtrd 2839	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
27		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
28	27, 11	mgpbas 20117	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
29		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝐾) = (._r‘𝐾)
30	27, 29	mgpplusg 20116	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝐾) = (+_g‘(mulGrp‘𝐾))
31	28, 30	sgrpcl 18685	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
32	25, 26, 7, 31	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
33	32, 6	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ 𝐵)
34	28, 30	sgrpcl 18685	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
35	25, 26, 9, 34	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
36	35, 6	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ 𝐵)
37	20	oveqrspc2v 7387	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑢(._r‘𝐾)𝑣) ∈ 𝐵 ∧ (𝑢(._r‘𝐾)𝑤) ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐾)𝑤)))
38	1, 33, 36, 37	syl12anc 837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐾)𝑤)))
39	17	oveqrspc2v 7387	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
40	39	ad2ant2r 748	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
41	17	oveqrspc2v 7387	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) = (𝑢(._r‘𝐿)𝑤))
42	1, 2, 8, 41	syl12anc 837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) = (𝑢(._r‘𝐿)𝑤))
43	40, 42	oveq12d 7378	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)))
44	38, 43	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)))
45	24, 44	eqeq12d 2753	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ↔ (𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤))))
46	11, 12	grpcl 18908	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
47	10, 46	syl3an1 1164	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
48	3, 26, 7, 47	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
49	48, 6	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ 𝐵)
50	17	oveqrspc2v 7387	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑢(+_g‘𝐾)𝑣) ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐿)𝑤))
51	1, 49, 8, 50	syl12anc 837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐿)𝑤))
52	20	oveqrspc2v 7387	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
53	52	ad2ant2r 748	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
54	53	oveq1d 7375	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐿)𝑤) = ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤))
55	51, 54	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤))
56	28, 30	sgrpcl 18685	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
57	25, 7, 9, 56	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
58	57, 6	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ 𝐵)
59	20	oveqrspc2v 7387	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑢(._r‘𝐾)𝑤) ∈ 𝐵 ∧ (𝑣(._r‘𝐾)𝑤) ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐾)𝑤)))
60	1, 36, 58, 59	syl12anc 837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐾)𝑤)))
61	17	oveqrspc2v 7387	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) = (𝑣(._r‘𝐿)𝑤))
62	1, 4, 8, 61	syl12anc 837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) = (𝑣(._r‘𝐿)𝑤))
63	42, 62	oveq12d 7378	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))
64	60, 63	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))
65	55, 64	eqeq12d 2753	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)) ↔ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))))
66	45, 65	anbi12d 633	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
67	66	anassrs 467	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
68	67	ralbidva 3159	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
69	68	2ralbidva 3200	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
70	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → 𝐵 = (Base‘𝐾))
71	70	raleqdv 3296	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))))
72	70, 71	raleqbidv 3312	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))))
73	70, 72	raleqbidv 3312	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))))
74		rngpropd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
75	74	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → 𝐵 = (Base‘𝐿))
76	75	raleqdv 3296	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))) ↔ ∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
77	75, 76	raleqbidv 3312	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
78	75, 77	raleqbidv 3312	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
79	69, 73, 78	3bitr3d 309	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
80	79	pm5.32da 579	. . . 4 ⊢ (𝜑 → (((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp) ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))) ↔ ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp) ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))))))
81		df-3an 1089	. . . 4 ⊢ ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))) ↔ ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp) ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))))
82		df-3an 1089	. . . 4 ⊢ ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))) ↔ ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp) ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
83	80, 81, 82	3bitr4g 314	. . 3 ⊢ (𝜑 → ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))) ↔ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))))))
84	5, 74, 20	ablpropd 19758	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
85		fvexd 6849	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ V)
86		fvexd 6849	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ V)
87	28	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
88		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
89		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘𝐿)
90	88, 89	mgpbas 20117	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
91	74, 90	eqtrdi 2788	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
92	5, 91	eqtr3d 2774	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐿)))
93	17	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝐿)𝑦)))
94	5	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
95	5	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
96	94, 95	anbi12d 633	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
97	96	bicomd 223	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
98	30	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝐾) = (+_g‘(mulGrp‘𝐾)))
99	98	eqcomd 2743	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾))
100	99	oveqd 7377	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(._r‘𝐾)𝑦))
101		eqid 2737	. . . . . . . . . . . 12 ⊢ (._r‘𝐿) = (._r‘𝐿)
102	88, 101	mgpplusg 20116	. . . . . . . . . . 11 ⊢ (._r‘𝐿) = (+_g‘(mulGrp‘𝐿))
103	102	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝐿) = (+_g‘(mulGrp‘𝐿)))
104	103	eqcomd 2743	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐿)) = (._r‘𝐿))
105	104	oveqd 7377	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐿))𝑦) = (𝑥(._r‘𝐿)𝑦))
106	100, 105	eqeq12d 2753	. . . . . . 7 ⊢ (𝜑 → ((𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(+_g‘(mulGrp‘𝐿))𝑦) ↔ (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝐿)𝑦)))
107	93, 97, 106	3imtr4d 294	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(+_g‘(mulGrp‘𝐿))𝑦)))
108	107	imp 406	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(+_g‘(mulGrp‘𝐿))𝑦))
109	85, 86, 87, 92, 108	sgrppropd 18690	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ Smgrp ↔ (mulGrp‘𝐿) ∈ Smgrp))
110	84, 109	3anbi12d 1440	. . 3 ⊢ (𝜑 → ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))) ↔ (𝐿 ∈ Abel ∧ (mulGrp‘𝐿) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))))))
111	83, 110	bitrd 279	. 2 ⊢ (𝜑 → ((𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))) ↔ (𝐿 ∈ Abel ∧ (mulGrp‘𝐿) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤))))))
112	11, 27, 12, 29	isrng 20126	. 2 ⊢ (𝐾 ∈ Rng ↔ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)∀𝑤 ∈ (Base‘𝐾)((𝑢(._r‘𝐾)(𝑣(+_g‘𝐾)𝑤)) = ((𝑢(._r‘𝐾)𝑣)(+_g‘𝐾)(𝑢(._r‘𝐾)𝑤)) ∧ ((𝑢(+_g‘𝐾)𝑣)(._r‘𝐾)𝑤) = ((𝑢(._r‘𝐾)𝑤)(+_g‘𝐾)(𝑣(._r‘𝐾)𝑤)))))
113		eqid 2737	. . 3 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
114	89, 88, 113, 101	isrng 20126	. 2 ⊢ (𝐿 ∈ Rng ↔ (𝐿 ∈ Abel ∧ (mulGrp‘𝐿) ∈ Smgrp ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)∀𝑤 ∈ (Base‘𝐿)((𝑢(._r‘𝐿)(𝑣(+_g‘𝐿)𝑤)) = ((𝑢(._r‘𝐿)𝑣)(+_g‘𝐿)(𝑢(._r‘𝐿)𝑤)) ∧ ((𝑢(+_g‘𝐿)𝑣)(._r‘𝐿)𝑤) = ((𝑢(._r‘𝐿)𝑤)(+_g‘𝐿)(𝑣(._r‘𝐿)𝑤)))))
115	111, 112, 114	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 +_gcplusg 17211 ._rcmulr 17212 Smgrpcsgrp 18677 Grpcgrp 18900 Abelcabl 19747 mulGrpcmgp 20112 Rngcrng 20124

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125

This theorem is referenced by: opprrngb 20317 subrngpropd 20536

W3C validator