rngpropd - Metamath Proof Explorer

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Theorem rngpropd 20090

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 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝜑)
2		simprll 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
3		simplrl 776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐾 ∈ Abel)
4		simprlr 779	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
7	4, 6	eleqtrd 2831	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ (Base‘𝐾))
8		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
9	8, 6	eleqtrd 2831	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		ablgrp 19722	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Abel → 𝐾 ∈ Grp)
11		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
13	11, 12	grpcl 18880	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
14	10, 13	syl3an1 1163	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
15	3, 7, 9, 14	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
16	15, 6	eleqtrrd 2832	. . . . . . . . . . . . 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 7417	. . . . . . . . . . . . 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 836	. . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
22	1, 4, 8, 21	syl12anc 836	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
23	22	oveq2d 7406	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Smgrp)
26	2, 6	eleqtrd 2831	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
27		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
28	27, 11	mgpbas 20061	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
29		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝐾) = (._r‘𝐾)
30	27, 29	mgpplusg 20060	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝐾) = (+_g‘(mulGrp‘𝐾))
31	28, 30	sgrpcl 18660	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
32	25, 26, 7, 31	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
33	32, 6	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ 𝐵)
34	28, 30	sgrpcl 18660	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
35	25, 26, 9, 34	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
36	35, 6	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ 𝐵)
37	20	oveqrspc2v 7417	. . . . . . . . . . . . 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 836	. . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
40	39	ad2ant2r 747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
41	17	oveqrspc2v 7417	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) = (𝑢(._r‘𝐿)𝑤))
42	1, 2, 8, 41	syl12anc 836	. . . . . . . . . . . . 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 7408	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 2746	. . . . . . . . . 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 18880	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
47	10, 46	syl3an1 1163	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
48	3, 26, 7, 47	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
49	48, 6	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ 𝐵)
50	17	oveqrspc2v 7417	. . . . . . . . . . . . 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 836	. . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
53	52	ad2ant2r 747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
54	53	oveq1d 7405	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 18660	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
57	25, 7, 9, 56	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
58	57, 6	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ 𝐵)
59	20	oveqrspc2v 7417	. . . . . . . . . . . . 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 836	. . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) = (𝑣(._r‘𝐿)𝑤))
62	1, 4, 8, 61	syl12anc 836	. . . . . . . . . . . . 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 7408	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 2746	. . . . . . . . . 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 632	. . . . . . . . 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 3155	. . . . . . 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 3301	. . . . . . . 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 3321	. . . . . . 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 3321	. . . . . 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 3301	. . . . . . . 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 3321	. . . . . . 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 3321	. . . . . 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 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 19729	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
85		fvexd 6876	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ V)
86		fvexd 6876	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ V)
87	28	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
88		eqid 2730	. . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
89		eqid 2730	. . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘𝐿)
90	88, 89	mgpbas 20061	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
91	74, 90	eqtrdi 2781	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
92	5, 91	eqtr3d 2767	. . . . 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 2815	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
95	5	eleq2d 2815	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
96	94, 95	anbi12d 632	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾))
100	99	oveqd 7407	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(._r‘𝐾)𝑦))
101		eqid 2730	. . . . . . . . . . . 12 ⊢ (._r‘𝐿) = (._r‘𝐿)
102	88, 101	mgpplusg 20060	. . . . . . . . . . 11 ⊢ (._r‘𝐿) = (+_g‘(mulGrp‘𝐿))
103	102	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝐿) = (+_g‘(mulGrp‘𝐿)))
104	103	eqcomd 2736	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐿)) = (._r‘𝐿))
105	104	oveqd 7407	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐿))𝑦) = (𝑥(._r‘𝐿)𝑦))
106	100, 105	eqeq12d 2746	. . . . . . 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 18665	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ Smgrp ↔ (mulGrp‘𝐿) ∈ Smgrp))
110	84, 109	3anbi12d 1439	. . 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 20070	. 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 2730	. . 3 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
114	89, 88, 113, 101	isrng 20070	. 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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 Vcvv 3450 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 ._rcmulr 17228 Smgrpcsgrp 18652 Grpcgrp 18872 Abelcabl 19718 mulGrpcmgp 20056 Rngcrng 20068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069

This theorem is referenced by: opprrngb 20262 subrngpropd 20484

W3C validator