rngpropd - Metamath Proof Explorer

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Theorem rngpropd 20083

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 2830	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		ablgrp 19715	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Abel → 𝐾 ∈ Grp)
11		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
13	11, 12	grpcl 18873	. . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 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 7403	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 2830	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
27		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
28	27, 11	mgpbas 20054	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
29		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝐾) = (._r‘𝐾)
30	27, 29	mgpplusg 20053	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝐾) = (+_g‘(mulGrp‘𝐾))
31	28, 30	sgrpcl 18653	. . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ 𝐵)
34	28, 30	sgrpcl 18653	. . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ 𝐵)
37	20	oveqrspc2v 7414	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 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 7405	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 2745	. . . . . . . . . 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 18873	. . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ 𝐵)
50	17	oveqrspc2v 7414	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 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 7402	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 18653	. . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ 𝐵)
59	20	oveqrspc2v 7414	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 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 7405	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 2745	. . . . . . . . . 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 3154	. . . . . . 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 3199	. . . . . 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 3299	. . . . . . . 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 3319	. . . . . . 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 3319	. . . . . 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 3299	. . . . . . . 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 3319	. . . . . . 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 3319	. . . . . 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 19722	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
85		fvexd 6873	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ V)
86		fvexd 6873	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ V)
87	28	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
88		eqid 2729	. . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
89		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘𝐿)
90	88, 89	mgpbas 20054	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
91	74, 90	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
92	5, 91	eqtr3d 2766	. . . . 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 2814	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
95	5	eleq2d 2814	. . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾))
100	99	oveqd 7404	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(._r‘𝐾)𝑦))
101		eqid 2729	. . . . . . . . . . . 12 ⊢ (._r‘𝐿) = (._r‘𝐿)
102	88, 101	mgpplusg 20053	. . . . . . . . . . 11 ⊢ (._r‘𝐿) = (+_g‘(mulGrp‘𝐿))
103	102	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝐿) = (+_g‘(mulGrp‘𝐿)))
104	103	eqcomd 2735	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐿)) = (._r‘𝐿))
105	104	oveqd 7404	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐿))𝑦) = (𝑥(._r‘𝐿)𝑦))
106	100, 105	eqeq12d 2745	. . . . . . 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 18658	. . . 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 20063	. 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 2729	. . 3 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
114	89, 88, 113, 101	isrng 20063	. 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 3044 Vcvv 3447 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 +_gcplusg 17220 ._rcmulr 17221 Smgrpcsgrp 18645 Grpcgrp 18865 Abelcabl 19711 mulGrpcmgp 20049 Rngcrng 20061

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 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062

This theorem is referenced by: opprrngb 20255 subrngpropd 20477

W3C validator