Theorem ringpropd 18405

Description: If two structures have the same group components (properties), 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 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝜑)
2		simprll 798	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ 𝐵)
3		simplrl 796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐾 ∈ Grp)
4		simprlr 799	. . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
7	4, 6	eleqtrd 2690	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑣 ∈ (Base‘𝐾))
8		simprr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
9	8, 6	eleqtrd 2690	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ (Base‘𝐾))
10		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝐾) = (+g‘𝐾)
12	10, 11	grpcl 17253	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(+g‘𝐾)𝑤) ∈ (Base‘𝐾))
13	3, 7, 9, 12	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) ∈ (Base‘𝐾))
14	13, 6	eleqtrrd 2691	. . . . . . . . . . . . 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 6572	. . . . . . . . . . . . 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 1316	. . . . . . . . . . . 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 6572	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
20	1, 4, 8, 19	syl12anc 1316	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
21	20	oveq2d 6565	. . . . . . . . . . . 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 2644	. . . . . . . . . . 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 797	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
24	2, 6	eleqtrd 2690	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → 𝑢 ∈ (Base‘𝐾))
25		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
26	25, 10	mgpbas 18318	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝐾) = (.r‘𝐾)
28	25, 27	mgpplusg 18316	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾))
29	26, 28	mndcl 17124	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(.r‘𝐾)𝑣) ∈ (Base‘𝐾))
30	23, 24, 7, 29	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) ∈ (Base‘𝐾))
31	30, 6	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) ∈ 𝐵)
32	26, 28	mndcl 17124	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑢(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
33	23, 24, 9, 32	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
34	33, 6	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) ∈ 𝐵)
35	18	oveqrspc2v 6572	. . . . . . . . . . . . 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 1316	. . . . . . . . . . . 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 6572	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) = (𝑢(.r‘𝐿)𝑣))
38	1, 2, 4, 37	syl12anc 1316	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) = (𝑢(.r‘𝐿)𝑣))
39	15	oveqrspc2v 6572	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) = (𝑢(.r‘𝐿)𝑤))
40	1, 2, 8, 39	syl12anc 1316	. . . . . . . . . . . . 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 6567	. . . . . . . . . . . 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 2644	. . . . . . . . . . 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 2625	. . . . . . . . . 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 17253	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐾) ∧ 𝑣 ∈ (Base‘𝐾)) → (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾))
45	3, 24, 7, 44	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾))
46	45, 6	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
47	15	oveqrspc2v 6572	. . . . . . . . . . . . 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 1316	. . . . . . . . . . . 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 6572	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
50	1, 2, 4, 49	syl12anc 1316	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
51	50	oveq1d 6564	. . . . . . . . . . . 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 2644	. . . . . . . . . . 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 17124	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑣 ∈ (Base‘𝐾) ∧ 𝑤 ∈ (Base‘𝐾)) → (𝑣(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
54	23, 7, 9, 53	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) ∈ (Base‘𝐾))
55	54, 6	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) ∈ 𝐵)
56	18	oveqrspc2v 6572	. . . . . . . . . . . . 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 1316	. . . . . . . . . . . 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 6572	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) = (𝑣(.r‘𝐿)𝑤))
59	1, 4, 8, 58	syl12anc 1316	. . . . . . . . . . . . 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 6567	. . . . . . . . . . . 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 2644	. . . . . . . . . . 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 2625	. . . . . . . . . 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 743	. . . . . . . . 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 678	. . . . . . . 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 2968	. . . . . . 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 2971	. . . . . 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 3121	. . . . . . . 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 3129	. . . . . . 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 3129	. . . . . 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 3121	. . . . . . . 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 3129	. . . . . . 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 3129	. . . . . 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 297	. . . . 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 671	. . . 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 1033	. . . 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 1033	. . . 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 302	. . 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 17260	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
82	5, 26	syl6eq 2660	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾)))
83		eqid 2610	. . . . . . 7 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿)
84		eqid 2610	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
85	83, 84	mgpbas 18318	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
86	71, 85	syl6eq 2660	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
87	28	oveqi 6562	. . . . . 6 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88		eqid 2610	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
89	83, 88	mgpplusg 18316	. . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿))
90	89	oveqi 6562	. . . . . 6 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)
91	15, 87, 90	3eqtr3g 2667	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦))
92	82, 86, 91	mndpropd 17139	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ Mnd ↔ (mulGrp‘𝐿) ∈ Mnd))
93	81, 92	3anbi12d 1392	. . 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 267	. 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 18374	. 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 2610	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
97	84, 83, 96, 88	isring 18374	. 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 302	1 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  Mndcmnd 17117  Grpcgrp 17245  mulGrpcmgp 18312  Ringcrg 18370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-mgp 18313  df-ring 18372

This theorem is referenced by:  crngpropd  18406  ringprop  18407  opprringb  18455  drngpropd  18597  subrgpropd  18637  rhmpropd  18638  abvpropd  18665  lmodprop2d  18748  sraassa  19146  assapropd  19148  subrgpsr  19240  opsrring  19436