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Theorem rngpropd 20203

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 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝜑)
2		simprll 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
3		simplrl 786	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐾 ∈ Abel)
4		simprlr 789	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
7	4, 6	eleqtrd 2863	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ (Base‘𝐾))
8		simprr 782	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
9	8, 6	eleqtrd 2863	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		ablgrp 19808	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Abel → 𝐾 ∈ Grp)
11		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
13	11, 12	grpcl 18966	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
14	10, 13	syl3an1 1175	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
15	3, 7, 9, 14	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) ∈ (Base‘𝐾))
16	15, 6	eleqtrrd 2864	. . . . . . . . . . . . 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 7419	. . . . . . . . . . . . 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 847	. . . . . . . . . . . 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 7419	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
22	1, 4, 8, 21	syl12anc 847	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+_g‘𝐾)𝑤) = (𝑣(+_g‘𝐿)𝑤))
23	22	oveq2d 7408	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 787	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Smgrp)
26	2, 6	eleqtrd 2863	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
27		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
28	27, 11	mgpbas 20174	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
29		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝐾) = (._r‘𝐾)
30	27, 29	mgpplusg 20173	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝐾) = (+_g‘(mulGrp‘𝐾))
31	28, 30	sgrpcl 18743	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
32	25, 26, 7, 31	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ (Base‘𝐾))
33	32, 6	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) ∈ 𝐵)
34	28, 30	sgrpcl 18743	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
35	25, 26, 9, 34	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
36	35, 6	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) ∈ 𝐵)
37	20	oveqrspc2v 7419	. . . . . . . . . . . . 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 847	. . . . . . . . . . . 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 7419	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
40	39	ad2ant2r 757	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑣) = (𝑢(._r‘𝐿)𝑣))
41	17	oveqrspc2v 7419	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝐾)𝑤) = (𝑢(._r‘𝐿)𝑤))
42	1, 2, 8, 41	syl12anc 847	. . . . . . . . . . . . 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 7410	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 2777	. . . . . . . . . 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 18966	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
47	10, 46	syl3an1 1175	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Abel ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
48	3, 26, 7, 47	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ (Base‘𝐾))
49	48, 6	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) ∈ 𝐵)
50	17	oveqrspc2v 7419	. . . . . . . . . . . . 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 847	. . . . . . . . . . . 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 7419	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
53	52	ad2ant2r 757	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+_g‘𝐾)𝑣) = (𝑢(+_g‘𝐿)𝑣))
54	53	oveq1d 7407	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 18743	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Smgrp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
57	25, 7, 9, 56	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ (Base‘𝐾))
58	57, 6	eleqtrrd 2864	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) ∈ 𝐵)
59	20	oveqrspc2v 7419	. . . . . . . . . . . . 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 847	. . . . . . . . . . . 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 7419	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(._r‘𝐾)𝑤) = (𝑣(._r‘𝐿)𝑤))
62	1, 4, 8, 61	syl12anc 847	. . . . . . . . . . . . 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 7410	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 2777	. . . . . . . . . 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 641	. . . . . . . . 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 471	. . . . . . . 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 3182	. . . . . . 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 3223	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → 𝐵 = (Base‘𝐾))
71	70	raleqdv 3319	. . . . . . . 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 3335	. . . . . . 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 3335	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Abel ∧ (mulGrp‘𝐾) ∈ Smgrp)) → 𝐵 = (Base‘𝐿))
76	75	raleqdv 3319	. . . . . . . 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 3335	. . . . . . 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 3335	. . . . . 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 311	. . . . 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 587	. . . 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 1099	. . . 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 1099	. . . 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 316	. . 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 19815	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
85		fvexd 6878	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ V)
86		fvexd 6878	. . . . 5 ⊢ (𝜑 → (mulGrp‘𝐿) ∈ V)
87	28	a1i 11	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
88		eqid 2761	. . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
89		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘𝐿)
90	88, 89	mgpbas 20174	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
91	74, 90	eqtrdi 2812	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
92	5, 91	eqtr3d 2798	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐿)))
93	17	ex 416	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(._r‘𝐾)𝑦) = (𝑥(._r‘𝐿)𝑦)))
94	5	eleq2d 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
95	5	eleq2d 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
96	94, 95	anbi12d 641	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
97	96	bicomd 225	. . . . . . 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 2767	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐾)) = (._r‘𝐾))
100	99	oveqd 7409	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(._r‘𝐾)𝑦))
101		eqid 2761	. . . . . . . . . . . 12 ⊢ (._r‘𝐿) = (._r‘𝐿)
102	88, 101	mgpplusg 20173	. . . . . . . . . . 11 ⊢ (._r‘𝐿) = (+_g‘(mulGrp‘𝐿))
103	102	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (._r‘𝐿) = (+_g‘(mulGrp‘𝐿)))
104	103	eqcomd 2767	. . . . . . . . 9 ⊢ (𝜑 → (+_g‘(mulGrp‘𝐿)) = (._r‘𝐿))
105	104	oveqd 7409	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+_g‘(mulGrp‘𝐿))𝑦) = (𝑥(._r‘𝐿)𝑦))
106	100, 105	eqeq12d 2777	. . . . . . 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 296	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥(+_g‘(mulGrp‘𝐾))𝑦) = (𝑥(+_g‘(mulGrp‘𝐿))𝑦)))
108	107	imp 410	. . . . 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 18748	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ Smgrp ↔ (mulGrp‘𝐿) ∈ Smgrp))
110	84, 109	3anbi12d 1457	. . 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 281	. 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 20183	. 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 2761	. . 3 ⊢ (+_g‘𝐿) = (+_g‘𝐿)
114	89, 88, 113, 101	isrng 20183	. 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 316	1 ⊢ (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 Vcvv 3453 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 +_gcplusg 17269 ._rcmulr 17270 Smgrpcsgrp 18735 Grpcgrp 18958 Abelcabl 19804 mulGrpcmgp 20169 Rngcrng 20181

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17282 df-0g 17453 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-grp 18961 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182

This theorem is referenced by: opprrngb 20374 subrngpropd 20597

W3C validator