Theorem iscringd 38509

Description: Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)

Hypotheses

Ref	Expression
iscringd.1	⊢ (𝜑 → 𝐺 ∈ AbelOp)
iscringd.2	⊢ (𝜑 → 𝑋 = ran 𝐺)
iscringd.3	⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋)
iscringd.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
iscringd.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
iscringd.6	⊢ (𝜑 → 𝑈 ∈ 𝑋)
iscringd.7	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦)
iscringd.8	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))

Assertion

Ref	Expression
iscringd	⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ CRingOps)

Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑈,𝑦

Allowed substitution hint:   𝑈(𝑧)

Proof of Theorem iscringd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscringd.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ AbelOp)
2		iscringd.2	. . 3 ⊢ (𝜑 → 𝑋 = ran 𝐺)
3		iscringd.3	. . 3 ⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋)
4		iscringd.4	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
5		iscringd.5	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
6		id 23	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋))
7	6	3com13 1140	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋))
8		eleq1 2853	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑋 ↔ 𝑧 ∈ 𝑋))
9	8	3anbi1d 1464	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) ↔ (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)))
10	9	anbi2d 641	. . . . . 6 ⊢ (𝑤 = 𝑧 → ((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋))))
11		oveq2 7408	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐺𝑦)𝐻𝑧))
12		oveq2 7408	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑥𝐻𝑤) = (𝑥𝐻𝑧))
13		oveq2 7408	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑦𝐻𝑤) = (𝑦𝐻𝑧))
14	12, 13	oveq12d 7418	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
15	11, 14	eqeq12d 2781	. . . . . 6 ⊢ (𝑤 = 𝑧 → (((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)) ↔ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
16	10, 15	imbi12d 347	. . . . 5 ⊢ (𝑤 = 𝑧 → (((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤))) ↔ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
17		eleq1 2853	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑋 ↔ 𝑥 ∈ 𝑋))
18	17	3anbi3d 1466	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ↔ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)))
19	18	anbi2d 641	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋))))
20		oveq1 7407	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝐺𝑦) = (𝑥𝐺𝑦))
21	20	oveq1d 7415	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑥𝐺𝑦)𝐻𝑤))
22		oveq1 7407	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧𝐻𝑤) = (𝑥𝐻𝑤))
23	22	oveq1d 7415	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)))
24	21, 23	eqeq12d 2781	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)) ↔ ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤))))
25	19, 24	imbi12d 347	. . . . . 6 ⊢ (𝑧 = 𝑥 → (((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤))) ↔ ((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)))))
26		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑋 ↔ 𝑤 ∈ 𝑋))
27	26	3anbi1d 1464	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ↔ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)))
28	27	anbi2d 641	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋))))
29		oveq2 7408	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐺𝑦)𝐻𝑤))
30		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑧𝐻𝑥) = (𝑧𝐻𝑤))
31		oveq2 7408	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑦𝐻𝑥) = (𝑦𝐻𝑤))
32	30, 31	oveq12d 7418	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)))
33	29, 32	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)) ↔ ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤))))
34	28, 33	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥))) ↔ ((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)))))
35	1	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐺 ∈ AbelOp)
36		simpr3 1213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
37	2	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑋 = ran 𝐺)
38	36, 37	eleqtrd 2867	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ ran 𝐺)
39		simpr2 1212	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
40	39, 37	eleqtrd 2867	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ ran 𝐺)
41		eqid 2765	. . . . . . . . . . 11 ⊢ ran 𝐺 = ran 𝐺
42	41	ablocom 30809	. . . . . . . . . 10 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) → (𝑧𝐺𝑦) = (𝑦𝐺𝑧))
43	35, 38, 40, 42	syl3anc 1394	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐺𝑦) = (𝑦𝐺𝑧))
44	43	oveq1d 7415	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑦𝐺𝑧)𝐻𝑥))
45		simpr1 1211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
46		ablogrpo 30808	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
47	35, 46	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐺 ∈ GrpOp)
48	41	grpocl 30761	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ ran 𝐺 ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐺𝑧) ∈ ran 𝐺)
49	47, 40, 38, 48	syl3anc 1394	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝐺𝑧) ∈ ran 𝐺)
50	49, 37	eleqtrrd 2868	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝐺𝑧) ∈ 𝑋)
51	45, 50	jca 520	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥 ∈ 𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋))
52		ovex 7433	. . . . . . . . . 10 ⊢ (𝑦𝐺𝑧) ∈ V
53		eleq1 2853	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑦𝐺𝑧) → (𝑤 ∈ 𝑋 ↔ (𝑦𝐺𝑧) ∈ 𝑋))
54	53	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑦𝐺𝑧) → ((𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋)))
55	54	anbi2d 641	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑦𝐺𝑧) → ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋))))
56		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑦𝐺𝑧) → (𝑥𝐻𝑤) = (𝑥𝐻(𝑦𝐺𝑧)))
57		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑦𝐺𝑧) → (𝑤𝐻𝑥) = ((𝑦𝐺𝑧)𝐻𝑥))
58	56, 57	eqeq12d 2781	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑦𝐺𝑧) → ((𝑥𝐻𝑤) = (𝑤𝐻𝑥) ↔ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥)))
59	55, 58	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑤 = (𝑦𝐺𝑧) → (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑥𝐻𝑤) = (𝑤𝐻𝑥)) ↔ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥))))
60		eleq1 2853	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝑋 ↔ 𝑤 ∈ 𝑋))
61	60	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)))
62	61	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋))))
63		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑥𝐻𝑦) = (𝑥𝐻𝑤))
64		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑦𝐻𝑥) = (𝑤𝐻𝑥))
65	63, 64	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥𝐻𝑤) = (𝑤𝐻𝑥)))
66	62, 65	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) ↔ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑥𝐻𝑤) = (𝑤𝐻𝑥))))
67		iscringd.8	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
68	66, 67	chvarvv 2012	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑥𝐻𝑤) = (𝑤𝐻𝑥))
69	52, 59, 68	vtocl 3528	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥))
70	51, 69	syldan 602	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥))
71	67	3adantr3 1188	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
72		eleq1 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑋 ↔ 𝑧 ∈ 𝑋))
73	72	anbi2d 641	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)))
74	73	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋))))
75		oveq2 7408	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑥𝐻𝑦) = (𝑥𝐻𝑧))
76		oveq1 7407	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑦𝐻𝑥) = (𝑧𝐻𝑥))
77	75, 76	eqeq12d 2781	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥𝐻𝑧) = (𝑧𝐻𝑥)))
78	74, 77	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) ↔ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))))
79	78, 67	chvarvv 2012	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))
80	79	3adantr2 1187	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))
81	71, 80	oveq12d 7418	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) = ((𝑦𝐻𝑥)𝐺(𝑧𝐻𝑥)))
82	3	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐻:(𝑋 × 𝑋)⟶𝑋)
83	82, 39, 45	fovcdmd 7572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝐻𝑥) ∈ 𝑋)
84	83, 37	eleqtrd 2867	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦𝐻𝑥) ∈ ran 𝐺)
85	82, 36, 45	fovcdmd 7572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐻𝑥) ∈ 𝑋)
86	85, 37	eleqtrd 2867	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧𝐻𝑥) ∈ ran 𝐺)
87	41	ablocom 30809	. . . . . . . . . 10 ⊢ ((𝐺 ∈ AbelOp ∧ (𝑦𝐻𝑥) ∈ ran 𝐺 ∧ (𝑧𝐻𝑥) ∈ ran 𝐺) → ((𝑦𝐻𝑥)𝐺(𝑧𝐻𝑥)) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
88	35, 84, 86, 87	syl3anc 1394	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦𝐻𝑥)𝐺(𝑧𝐻𝑥)) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
89	5, 81, 88	3eqtrd 2804	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
90	44, 70, 89	3eqtr2d 2806	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
91	34, 90	chvarvv 2012	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)))
92	25, 91	chvarvv 2012	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)))
93	16, 92	chvarvv 2012	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
94	7, 93	sylan2 604	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
95		iscringd.6	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑋)
96	95	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ∈ 𝑋)
97		oveq1 7407	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
98		oveq2 7408	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → (𝑦𝐻𝑥) = (𝑦𝐻𝑈))
99	97, 98	eqeq12d 2781	. . . . . . 7 ⊢ (𝑥 = 𝑈 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑈𝐻𝑦) = (𝑦𝐻𝑈)))
100	99	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝑈 → (((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = (𝑦𝐻𝑈))))
101	67	an12s 661	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝜑 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
102	101	ex 417	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
103	100, 102	vtoclga 3544	. . . . 5 ⊢ (𝑈 ∈ 𝑋 → ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = (𝑦𝐻𝑈)))
104	96, 103	mpcom 39	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = (𝑦𝐻𝑈))
105		iscringd.7	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦)
106	104, 105	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = 𝑦)
107	1, 2, 3, 4, 5, 94, 95, 106, 105	isrngod 38409	. 2 ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)
108	2	eleq2d 2851	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ran 𝐺))
109	2	eleq2d 2851	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ran 𝐺))
110	108, 109	anbi12d 643	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)))
111	110	biimpar 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
112	111, 67	syldan 602	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
113	112	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐻𝑦) = (𝑦𝐻𝑥))
114		rnexg 7887	. . . . . . . 8 ⊢ (𝐺 ∈ AbelOp → ran 𝐺 ∈ V)
115	1, 114	syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ∈ V)
116	2, 115	eqeltrd 2865	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
117	116, 116	xpexd 7738	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑋) ∈ V)
118	3, 117	fexd 7215	. . . 4 ⊢ (𝜑 → 𝐻 ∈ V)
119		iscom2 38506	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐻 ∈ V) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
120	1, 118, 119	syl2anc 595	. . 3 ⊢ (𝜑 → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
121	113, 120	mpbird 260	. 2 ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ Com2)
122		iscrngo 38507	. 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ CRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ ⟨𝐺, 𝐻⟩ ∈ Com2))
123	107, 121, 122	sylanbrc 594	1 ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ CRingOps)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457  ⟨cop 4591   × cxp 5650  ran crn 5653  ⟶wf 6521  (class class class)co 7400  GrpOpcgr 30750  AbelOpcablo 30805  RingOpscrngo 38405  Com2ccm2 38500  CRingOpsccring 38504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-grpo 30754  df-ablo 30806  df-rngo 38406  df-com2 38501  df-crngo 38505

This theorem is referenced by: (None)