Theorem ringpropd 20253

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 2837	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . 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 18929	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . 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 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (mulGrp‘𝐾) ∈ Mnd)
24	2, 6	eleqtrd 2837	. . . . . . . . . . . . . . 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 20110	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
27		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝐾) = (.r‘𝐾)
28	25, 27	mgpplusg 20109	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾))
29	26, 28	mndcl 18725	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑣) ∈ 𝐵)
32	26, 28	mndcl 18725	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝐾)𝑤) ∈ 𝐵)
35	18	oveqrspc2v 7437	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . 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 18929	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
47	15	oveqrspc2v 7437	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . 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 7425	. . . . . . . . . . . 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 18725	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐾 ∈ Grp ∧ (mulGrp‘𝐾) ∈ Mnd)) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵)) → (𝑣(.r‘𝐾)𝑤) ∈ 𝐵)
56	18	oveqrspc2v 7437	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . 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 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 3162	. . . . . . 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 3207	. . . . . 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 3309	. . . . . . . 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 3329	. . . . . . 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 3329	. . . . . 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 3309	. . . . . . . 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 3329	. . . . . . 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 3329	. . . . . 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 18939	. . . 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 20110	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿))
86	71, 85	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿)))
87	28	oveqi 7423	. . . . . 6 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)
88		eqid 2736	. . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿)
89	83, 88	mgpplusg 20109	. . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿))
90	89	oveqi 7423	. . . . . 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 18742	. . . 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 20202	. 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 20202	. 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 3052  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  ._rcmulr 17277  Mndcmnd 18717  Grpcgrp 18921  mulGrpcmgp 20105  Ringcrg 20198

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-plusg 17289  df-0g 17460  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-mgp 20106  df-ring 20200

This theorem is referenced by:  crngpropd  20254  ringprop  20255  opprringb  20313  nzrpropd  20485  subrgpropd  20573  rhmpropd  20574  drngpropd  20734  abvpropd  20800  lmodprop2d  20886  sraring  21149  sraassaOLD  21835  assapropd  21837  subrgpsr  21943  opsrring  22185