rngpropd - Metamath Proof Explorer

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Theorem rngpropd 20069

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 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝜑)
2		simprll 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
3		simplrl 774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐾 ∈ Abel)
4		simprlr 777	. . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
7	4, 6	eleqtrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ (Base‘𝐾))
8		simprr 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
9	8, 6	eleqtrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		ablgrp 19695	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Abel → 𝐾 ∈ Grp)
11		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
13	11, 12	grpcl 18864	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
14	10, 13	syl3an1 1162	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
15	3, 7, 9, 14	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
16	15, 6	eleqtrrd 2835	. . . . . . . . . . . . 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 7439	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
22	1, 4, 8, 21	syl12anc 834	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
23	22	oveq2d 7428	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Smgrp)
26	2, 6	eleqtrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
27		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
28	27, 11	mgpbas 20035	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
29		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝐾) = (._r‘𝐾)
30	27, 29	mgpplusg 20033	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝐾) = (+_g‘(mulGrp‘𝐾))
31	28, 30	sgrpcl 18652	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
32	25, 26, 7, 31	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
33	32, 6	eleqtrrd 2835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ 𝐵)
34	28, 30	sgrpcl 18652	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
35	25, 26, 9, 34	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
36	35, 6	eleqtrrd 2835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ 𝐵)
37	20	oveqrspc2v 7439	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
40	39	ad2ant2r 744	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
41	17	oveqrspc2v 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) = (𝑢(._r‘𝐿)𝑤))
42	1, 2, 8, 41	syl12anc 834	. . . . . . . . . . . . 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 7430	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2747	. . . . . . . . . 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 18864	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
47	10, 46	syl3an1 1162	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
48	3, 26, 7, 47	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
49	48, 6	eleqtrrd 2835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ 𝐵)
50	17	oveqrspc2v 7439	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
53	52	ad2ant2r 744	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
54	53	oveq1d 7427	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 18652	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
57	25, 7, 9, 56	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
58	57, 6	eleqtrrd 2835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ 𝐵)
59	20	oveqrspc2v 7439	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) = (𝑣(._r‘𝐿)𝑤))
62	1, 4, 8, 61	syl12anc 834	. . . . . . . . . . . . 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 7430	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2747	. . . . . . . . . 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 630	. . . . . . . . 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 3174	. . . . . . 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 3215	. . . . . 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 3324	. . . . . . . 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 3341	. . . . . . 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 3341	. . . . . 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 3324	. . . . . . . 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 3341	. . . . . . 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 3341	. . . . . 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 578	. . . 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 1088	. . . 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 1088	. . . 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 19702	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
85		fvexd 6906	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ V)
86		fvexd 6906	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ V)
87	28	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
88		eqid 2731	. . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
89		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘𝐿)
90	88, 89	mgpbas 20035	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
91	74, 90	eqtrdi 2787	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
92	5, 91	eqtr3d 2773	. . . . 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 2818	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
95	5	eleq2d 2818	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
96	94, 95	anbi12d 630	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
97	96	bicomd 222	. . . . . . 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 2737	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾))
100	99	oveqd 7429	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(._r‘𝐾)𝑦))
101		eqid 2731	. . . . . . . . . . . 12 ⊢ (._r‘𝐿) = (._r‘𝐿)
102	88, 101	mgpplusg 20033	. . . . . . . . . . 11 ⊢ (._r‘𝐿) = (+_g‘(mulGrp‘𝐿))
103	102	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝐿) = (+_g‘(mulGrp‘𝐿)))
104	103	eqcomd 2737	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐿)) = (._r‘𝐿))
105	104	oveqd 7429	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐿))𝑦) = (𝑥(._r‘𝐿)𝑦))
106	100, 105	eqeq12d 2747	. . . . . . 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 18657	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ Smgrp ↔ (mulGrp‘𝐿) ∈ Smgrp))
110	84, 109	3anbi12d 1436	. . 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 20049	. 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 2731	. . 3 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
114	89, 88, 113, 101	isrng 20049	. 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 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 +_gcplusg 17202 ._rcmulr 17203 Smgrpcsgrp 18644 Grpcgrp 18856 Abelcabl 19691 mulGrpcmgp 20029 Rngcrng 20047

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048

This theorem is referenced by: opprrngb 20238 subrngpropd 20457

W3C validator