Theorem ringpropd 20197

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 ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ringpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
ringpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
ringpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
ringpropd.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))

Assertion

Ref	Expression
ringpropd	⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))

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

Proof of Theorem ringpropd

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝜑)
2		simprll 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
3		simplrl 776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐾 ∈ Grp)
4		simprlr 779	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
5		ringpropd.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
6	5	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
7	4, 6	eleqtrd 2830	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ (Base‘𝐾))
8		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
9	8, 6	eleqtrd 2830	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝐾) = (+g‘𝐾)
12	10, 11	grpcl 18873	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+g‘𝐾)𝑤) ∈ (Base‘𝐾))
13	3, 7, 9, 12	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) ∈ (Base‘𝐾))
14	13, 6	eleqtrrd 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) ∈ 𝐵)
15		ringpropd.4	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
16	15	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)))
17	1, 2, 14, 16	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(.r‘𝐿)(𝑣(+g‘𝐾)𝑤)))
18		ringpropd.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
19	18	oveqrspc2v 7414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
20	1, 4, 8, 19	syl12anc 836	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
21	20	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐿)(𝑣(+g‘𝐾)𝑤)) = (𝑢(.r‘𝐿)(𝑣(+g‘𝐿)𝑤)))
22	17, 21	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(.r‘𝐿)(𝑣(+g‘𝐿)𝑤)))
23		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
24	2, 6	eleqtrd 2830	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
25		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
26	25, 10	mgpbas 20054	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝐾) = (.r‘𝐾)
28	25, 27	mgpplusg 20053	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾))
29	26, 28	mndcl 18669	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(.r‘𝐾)𝑣) ∈ (Base‘𝐾))
30	23, 24, 7, 29	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) ∈ (Base‘𝐾))
31	30, 6	eleqtrrd 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) ∈ 𝐵)
32	26, 28	mndcl 18669	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
33	23, 24, 9, 32	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
34	33, 6	eleqtrrd 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) ∈ 𝐵)
35	18	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)))
36	1, 31, 34, 35	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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)))
37	15	oveqrspc2v 7414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) = (𝑢(.r‘𝐿)𝑣))
38	1, 2, 4, 37	syl12anc 836	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) = (𝑢(.r‘𝐿)𝑣))
39	15	oveqrspc2v 7414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) = (𝑢(.r‘𝐿)𝑤))
40	1, 2, 8, 39	syl12anc 836	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) = (𝑢(.r‘𝐿)𝑤))
41	38, 40	oveq12d 7405	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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)))
42	36, 41	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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)))
43	22, 42	eqeq12d 2745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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))))
44	10, 11	grpcl 18873	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾))
45	3, 24, 7, 44	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾))
46	45, 6	eleqtrrd 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
47	15	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))
48	1, 46, 8, 47	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣)(.r‘𝐾)𝑤) = ((𝑢(+g‘𝐾)𝑣)(.r‘𝐿)𝑤))
49	18	oveqrspc2v 7414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
50	1, 2, 4, 49	syl12anc 836	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
51	50	oveq1d 7402	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣)(.r‘𝐿)𝑤) = ((𝑢(+g‘𝐿)𝑣)(.r‘𝐿)𝑤))
52	48, 51	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣)(.r‘𝐾)𝑤) = ((𝑢(+g‘𝐿)𝑣)(.r‘𝐿)𝑤))
53	26, 28	mndcl 18669	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
54	23, 7, 9, 53	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
55	54, 6	eleqtrrd 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) ∈ 𝐵)
56	18	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)))
57	1, 34, 55, 56	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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)))
58	15	oveqrspc2v 7414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) = (𝑣(.r‘𝐿)𝑤))
59	1, 4, 8, 58	syl12anc 836	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) = (𝑣(.r‘𝐿)𝑤))
60	40, 59	oveq12d 7405	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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)))
61	57, 60	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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)))
62	52, 61	eqeq12d 2745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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))))
63	43, 62	anbi12d 632	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((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)))))
64	63	anassrs 467	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ (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)))))
65	64	ralbidva 3154	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ (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)))))
66	65	2ralbidva 3199	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
67	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐾))
68	67	raleqdv 3299	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
69	67, 68	raleqbidv 3319	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
70	67, 69	raleqbidv 3319	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
71		ringpropd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
72	71	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → 𝐵 = (Base‘𝐿))
73	72	raleqdv 3299	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
74	72, 73	raleqbidv 3319	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
75	72, 74	raleqbidv 3319	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
76	66, 70, 75	3bitr3d 309	. . . . 5 ⊢ ((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) → (∀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)))))
77	76	pm5.32da 579	. . . 4 ⊢ (𝜑 → (((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀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 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀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))))))
78		df-3an 1088	. . . 4 ⊢ ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀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 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀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)))))
79		df-3an 1088	. . . 4 ⊢ ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀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 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd) ∧ ∀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	77, 78, 79	3bitr4g 314	. . 3 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀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 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀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	5, 71, 18	grppropd 18883	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
82	5, 26	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾)))
83		eqid 2729	. . . . . . 7 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
84		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
85	83, 84	mgpbas 20054	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
86	71, 85	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
87	28	oveqi 7400	. . . . . 6 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88		eqid 2729	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
89	83, 88	mgpplusg 20053	. . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿))
90	89	oveqi 7400	. . . . . 6 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
91	15, 87, 90	3eqtr3g 2787	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
92	82, 86, 91	mndpropd 18686	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ Mnd ↔ (mulGrp‘𝐿) ∈ Mnd))
93	81, 92	3anbi12d 1439	. . 3 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀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 ∈ Grp ∧ (mulGrp‘𝐿) ∈ Mnd ∧ ∀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))))))
94	80, 93	bitrd 279	. 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀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 ∈ Grp ∧ (mulGrp‘𝐿) ∈ Mnd ∧ ∀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))))))
95	10, 25, 11, 27	isring 20146	. 2 ⊢ (𝐾 ∈ Ring ↔ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd ∧ ∀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)))))
96		eqid 2729	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
97	84, 83, 96, 88	isring 20146	. 2 ⊢ (𝐿 ∈ Ring ↔ (𝐿 ∈ Grp ∧ (mulGrp‘𝐿) ∈ Mnd ∧ ∀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)))))
98	94, 95, 97	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Mndcmnd 18661  Grpcgrp 18865  mulGrpcmgp 20049  Ringcrg 20142

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-mgp 20050  df-ring 20144

This theorem is referenced by:  crngpropd  20198  ringprop  20199  opprringb  20257  nzrpropd  20429  subrgpropd  20517  rhmpropd  20518  drngpropd  20678  abvpropd  20744  lmodprop2d  20830  sraring  21093  sraassaOLD  21779  assapropd  21781  subrgpsr  21887  opsrring  22129