Theorem ringpropd 20269

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 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝜑)
2		simprll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
3		simplrl 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐾 ∈ Grp)
4		simprlr 780	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
7	4, 6	eleqtrd 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ (Base‘𝐾))
8		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
9	8, 6	eleqtrd 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝐾) = (+g‘𝐾)
12	10, 11	grpcl 18917	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+g‘𝐾)𝑤) ∈ (Base‘𝐾))
13	3, 7, 9, 12	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) ∈ (Base‘𝐾))
14	13, 6	eleqtrrd 2839	. . . . . . . . . . . . 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 7394	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 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 7394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
20	1, 4, 8, 19	syl12anc 837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
21	20	oveq2d 7383	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
24	2, 6	eleqtrd 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
25		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
26	25, 10	mgpbas 20126	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝐾) = (.r‘𝐾)
28	25, 27	mgpplusg 20125	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾))
29	26, 28	mndcl 18710	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(.r‘𝐾)𝑣) ∈ (Base‘𝐾))
30	23, 24, 7, 29	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) ∈ (Base‘𝐾))
31	30, 6	eleqtrrd 2839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) ∈ 𝐵)
32	26, 28	mndcl 18710	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
33	23, 24, 9, 32	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
34	33, 6	eleqtrrd 2839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) ∈ 𝐵)
35	18	oveqrspc2v 7394	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 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 7394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) = (𝑢(.r‘𝐿)𝑣))
38	1, 2, 4, 37	syl12anc 837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) = (𝑢(.r‘𝐿)𝑣))
39	15	oveqrspc2v 7394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) = (𝑢(.r‘𝐿)𝑤))
40	1, 2, 8, 39	syl12anc 837	. . . . . . . . . . . . 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 7385	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2752	. . . . . . . . . 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 18917	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾))
45	3, 24, 7, 44	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾))
46	45, 6	eleqtrrd 2839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
47	15	oveqrspc2v 7394	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 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 7394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
50	1, 2, 4, 49	syl12anc 837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
51	50	oveq1d 7382	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 18710	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
54	23, 7, 9, 53	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
55	54, 6	eleqtrrd 2839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) ∈ 𝐵)
56	18	oveqrspc2v 7394	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 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 7394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) = (𝑣(.r‘𝐿)𝑤))
59	1, 4, 8, 58	syl12anc 837	. . . . . . . . . . . . 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 7385	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2752	. . . . . . . . . 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 633	. . . . . . . . 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 3158	. . . . . . 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 3295	. . . . . . . 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 3311	. . . . . . 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 3311	. . . . . 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 3295	. . . . . . . 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 3311	. . . . . . 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 3311	. . . . . 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 1089	. . . 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 1089	. . . 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 18927	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
82	5, 26	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾)))
83		eqid 2736	. . . . . . 7 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
84		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
85	83, 84	mgpbas 20126	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
86	71, 85	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
87	28	oveqi 7380	. . . . . 6 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88		eqid 2736	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
89	83, 88	mgpplusg 20125	. . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿))
90	89	oveqi 7380	. . . . . 6 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
91	15, 87, 90	3eqtr3g 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
92	82, 86, 91	mndpropd 18727	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ Mnd ↔ (mulGrp‘𝐿) ∈ Mnd))
93	81, 92	3anbi12d 1440	. . 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 20218	. 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 2736	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
97	84, 83, 96, 88	isring 20218	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Mndcmnd 18702  Grpcgrp 18909  mulGrpcmgp 20121  Ringcrg 20214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-mgp 20122  df-ring 20216

This theorem is referenced by:  crngpropd  20270  ringprop  20271  opprringb  20328  nzrpropd  20497  subrgpropd  20585  rhmpropd  20586  drngpropd  20746  abvpropd  20812  lmodprop2d  20919  sraring  21181  assapropd  21851  subrgpsr  21956  opsrring  22208