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Theorem iscringd 34106
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 (𝜑𝐺 ∈ AbelOp)
iscringd.2 (𝜑𝑋 = ran 𝐺)
iscringd.3 (𝜑𝐻:(𝑋 × 𝑋)⟶𝑋)
iscringd.4 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
iscringd.5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
iscringd.6 (𝜑𝑈𝑋)
iscringd.7 ((𝜑𝑦𝑋) → (𝑦𝐻𝑈) = 𝑦)
iscringd.8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (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.
StepHypRef Expression
1 iscringd.1 . . 3 (𝜑𝐺 ∈ AbelOp)
2 iscringd.2 . . 3 (𝜑𝑋 = ran 𝐺)
3 iscringd.3 . . 3 (𝜑𝐻:(𝑋 × 𝑋)⟶𝑋)
4 iscringd.4 . . 3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
5 iscringd.5 . . 3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
6 id 22 . . . . 5 ((𝑧𝑋𝑦𝑋𝑥𝑋) → (𝑧𝑋𝑦𝑋𝑥𝑋))
763com13 1147 . . . 4 ((𝑥𝑋𝑦𝑋𝑧𝑋) → (𝑧𝑋𝑦𝑋𝑥𝑋))
8 eleq1 2873 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝑋𝑧𝑋))
983anbi1d 1557 . . . . . . 7 (𝑤 = 𝑧 → ((𝑤𝑋𝑦𝑋𝑥𝑋) ↔ (𝑧𝑋𝑦𝑋𝑥𝑋)))
109anbi2d 616 . . . . . 6 (𝑤 = 𝑧 → ((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑥𝑋)) ↔ (𝜑 ∧ (𝑧𝑋𝑦𝑋𝑥𝑋))))
11 oveq2 6878 . . . . . . 7 (𝑤 = 𝑧 → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐺𝑦)𝐻𝑧))
12 oveq2 6878 . . . . . . . 8 (𝑤 = 𝑧 → (𝑥𝐻𝑤) = (𝑥𝐻𝑧))
13 oveq2 6878 . . . . . . . 8 (𝑤 = 𝑧 → (𝑦𝐻𝑤) = (𝑦𝐻𝑧))
1412, 13oveq12d 6888 . . . . . . 7 (𝑤 = 𝑧 → ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
1511, 14eqeq12d 2821 . . . . . 6 (𝑤 = 𝑧 → (((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)) ↔ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
1610, 15imbi12d 335 . . . . 5 (𝑤 = 𝑧 → (((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑥𝑋)) → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤))) ↔ ((𝜑 ∧ (𝑧𝑋𝑦𝑋𝑥𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
17 eleq1 2873 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧𝑋𝑥𝑋))
18173anbi3d 1559 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑤𝑋𝑦𝑋𝑧𝑋) ↔ (𝑤𝑋𝑦𝑋𝑥𝑋)))
1918anbi2d 616 . . . . . . 7 (𝑧 = 𝑥 → ((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑧𝑋)) ↔ (𝜑 ∧ (𝑤𝑋𝑦𝑋𝑥𝑋))))
20 oveq1 6877 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧𝐺𝑦) = (𝑥𝐺𝑦))
2120oveq1d 6885 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑥𝐺𝑦)𝐻𝑤))
22 oveq1 6877 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧𝐻𝑤) = (𝑥𝐻𝑤))
2322oveq1d 6885 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)))
2421, 23eqeq12d 2821 . . . . . . 7 (𝑧 = 𝑥 → (((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)) ↔ ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤))))
2519, 24imbi12d 335 . . . . . 6 (𝑧 = 𝑥 → (((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑧𝑋)) → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤))) ↔ ((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑥𝑋)) → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)))))
26 eleq1 2873 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑥𝑋𝑤𝑋))
27263anbi1d 1557 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑥𝑋𝑦𝑋𝑧𝑋) ↔ (𝑤𝑋𝑦𝑋𝑧𝑋)))
2827anbi2d 616 . . . . . . . 8 (𝑥 = 𝑤 → ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) ↔ (𝜑 ∧ (𝑤𝑋𝑦𝑋𝑧𝑋))))
29 oveq2 6878 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐺𝑦)𝐻𝑤))
30 oveq2 6878 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑧𝐻𝑥) = (𝑧𝐻𝑤))
31 oveq2 6878 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑦𝐻𝑥) = (𝑦𝐻𝑤))
3230, 31oveq12d 6888 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)))
3329, 32eqeq12d 2821 . . . . . . . 8 (𝑥 = 𝑤 → (((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)) ↔ ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤))))
3428, 33imbi12d 335 . . . . . . 7 (𝑥 = 𝑤 → (((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥))) ↔ ((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑧𝑋)) → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)))))
351adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝐺 ∈ AbelOp)
36 simpr3 1245 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
372adantr 468 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝑋 = ran 𝐺)
3836, 37eleqtrd 2887 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝑧 ∈ ran 𝐺)
39 simpr2 1243 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
4039, 37eleqtrd 2887 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝑦 ∈ ran 𝐺)
41 eqid 2806 . . . . . . . . . . 11 ran 𝐺 = ran 𝐺
4241ablocom 27730 . . . . . . . . . 10 ((𝐺 ∈ AbelOp ∧ 𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐺) → (𝑧𝐺𝑦) = (𝑦𝐺𝑧))
4335, 38, 40, 42syl3anc 1483 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑧𝐺𝑦) = (𝑦𝐺𝑧))
4443oveq1d 6885 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑦𝐺𝑧)𝐻𝑥))
45 simpr1 1241 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝑥𝑋)
46 ablogrpo 27729 . . . . . . . . . . . . 13 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4735, 46syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝐺 ∈ GrpOp)
4841grpocl 27682 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺) → (𝑦𝐺𝑧) ∈ ran 𝐺)
4947, 40, 38, 48syl3anc 1483 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑦𝐺𝑧) ∈ ran 𝐺)
5049, 37eleqtrrd 2888 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑦𝐺𝑧) ∈ 𝑋)
5145, 50jca 503 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋))
52 ovex 6902 . . . . . . . . . 10 (𝑦𝐺𝑧) ∈ V
53 eleq1 2873 . . . . . . . . . . . . 13 (𝑤 = (𝑦𝐺𝑧) → (𝑤𝑋 ↔ (𝑦𝐺𝑧) ∈ 𝑋))
5453anbi2d 616 . . . . . . . . . . . 12 (𝑤 = (𝑦𝐺𝑧) → ((𝑥𝑋𝑤𝑋) ↔ (𝑥𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋)))
5554anbi2d 616 . . . . . . . . . . 11 (𝑤 = (𝑦𝐺𝑧) → ((𝜑 ∧ (𝑥𝑋𝑤𝑋)) ↔ (𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋))))
56 oveq2 6878 . . . . . . . . . . . 12 (𝑤 = (𝑦𝐺𝑧) → (𝑥𝐻𝑤) = (𝑥𝐻(𝑦𝐺𝑧)))
57 oveq1 6877 . . . . . . . . . . . 12 (𝑤 = (𝑦𝐺𝑧) → (𝑤𝐻𝑥) = ((𝑦𝐺𝑧)𝐻𝑥))
5856, 57eqeq12d 2821 . . . . . . . . . . 11 (𝑤 = (𝑦𝐺𝑧) → ((𝑥𝐻𝑤) = (𝑤𝐻𝑥) ↔ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥)))
5955, 58imbi12d 335 . . . . . . . . . 10 (𝑤 = (𝑦𝐺𝑧) → (((𝜑 ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐻𝑤) = (𝑤𝐻𝑥)) ↔ ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥))))
60 eleq1 2873 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝑦𝑋𝑤𝑋))
6160anbi2d 616 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝑥𝑋𝑦𝑋) ↔ (𝑥𝑋𝑤𝑋)))
6261anbi2d 616 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) ↔ (𝜑 ∧ (𝑥𝑋𝑤𝑋))))
63 oveq2 6878 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑥𝐻𝑦) = (𝑥𝐻𝑤))
64 oveq1 6877 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑦𝐻𝑥) = (𝑤𝐻𝑥))
6563, 64eqeq12d 2821 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥𝐻𝑤) = (𝑤𝐻𝑥)))
6662, 65imbi12d 335 . . . . . . . . . . 11 (𝑦 = 𝑤 → (((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) ↔ ((𝜑 ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐻𝑤) = (𝑤𝐻𝑥))))
67 iscringd.8 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
6866, 67chvarv 2437 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐻𝑤) = (𝑤𝐻𝑥))
6952, 59, 68vtocl 3452 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝐺𝑧) ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥))
7051, 69syldan 581 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑦𝐺𝑧)𝐻𝑥))
71673adantr3 1205 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
72 eleq1 2873 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → (𝑦𝑋𝑧𝑋))
7372anbi2d 616 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ((𝑥𝑋𝑦𝑋) ↔ (𝑥𝑋𝑧𝑋)))
7473anbi2d 616 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) ↔ (𝜑 ∧ (𝑥𝑋𝑧𝑋))))
75 oveq2 6878 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑥𝐻𝑦) = (𝑥𝐻𝑧))
76 oveq1 6877 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝑦𝐻𝑥) = (𝑧𝐻𝑥))
7775, 76eqeq12d 2821 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑥𝐻𝑧) = (𝑧𝐻𝑥)))
7874, 77imbi12d 335 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) ↔ ((𝜑 ∧ (𝑥𝑋𝑧𝑋)) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))))
7978, 67chvarv 2437 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑧𝑋)) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))
80793adantr2 1204 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻𝑧) = (𝑧𝐻𝑥))
8171, 80oveq12d 6888 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) = ((𝑦𝐻𝑥)𝐺(𝑧𝐻𝑥)))
823adantr 468 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → 𝐻:(𝑋 × 𝑋)⟶𝑋)
8382, 39, 45fovrnd 7032 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑦𝐻𝑥) ∈ 𝑋)
8483, 37eleqtrd 2887 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑦𝐻𝑥) ∈ ran 𝐺)
8582, 36, 45fovrnd 7032 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑧𝐻𝑥) ∈ 𝑋)
8685, 37eleqtrd 2887 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑧𝐻𝑥) ∈ ran 𝐺)
8741ablocom 27730 . . . . . . . . . 10 ((𝐺 ∈ AbelOp ∧ (𝑦𝐻𝑥) ∈ ran 𝐺 ∧ (𝑧𝐻𝑥) ∈ ran 𝐺) → ((𝑦𝐻𝑥)𝐺(𝑧𝐻𝑥)) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
8835, 84, 86, 87syl3anc 1483 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑦𝐻𝑥)𝐺(𝑧𝐻𝑥)) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
895, 81, 883eqtrd 2844 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
9044, 70, 893eqtr2d 2846 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑧𝐺𝑦)𝐻𝑥) = ((𝑧𝐻𝑥)𝐺(𝑦𝐻𝑥)))
9134, 90chvarv 2437 . . . . . 6 ((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑧𝑋)) → ((𝑧𝐺𝑦)𝐻𝑤) = ((𝑧𝐻𝑤)𝐺(𝑦𝐻𝑤)))
9225, 91chvarv 2437 . . . . 5 ((𝜑 ∧ (𝑤𝑋𝑦𝑋𝑥𝑋)) → ((𝑥𝐺𝑦)𝐻𝑤) = ((𝑥𝐻𝑤)𝐺(𝑦𝐻𝑤)))
9316, 92chvarv 2437 . . . 4 ((𝜑 ∧ (𝑧𝑋𝑦𝑋𝑥𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
947, 93sylan2 582 . . 3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
95 iscringd.6 . . 3 (𝜑𝑈𝑋)
9695adantr 468 . . . . 5 ((𝜑𝑦𝑋) → 𝑈𝑋)
97 oveq1 6877 . . . . . . . 8 (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
98 oveq2 6878 . . . . . . . 8 (𝑥 = 𝑈 → (𝑦𝐻𝑥) = (𝑦𝐻𝑈))
9997, 98eqeq12d 2821 . . . . . . 7 (𝑥 = 𝑈 → ((𝑥𝐻𝑦) = (𝑦𝐻𝑥) ↔ (𝑈𝐻𝑦) = (𝑦𝐻𝑈)))
10099imbi2d 331 . . . . . 6 (𝑥 = 𝑈 → (((𝜑𝑦𝑋) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)) ↔ ((𝜑𝑦𝑋) → (𝑈𝐻𝑦) = (𝑦𝐻𝑈))))
10167an12s 631 . . . . . . 7 ((𝑥𝑋 ∧ (𝜑𝑦𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
102101ex 399 . . . . . 6 (𝑥𝑋 → ((𝜑𝑦𝑋) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
103100, 102vtoclga 3465 . . . . 5 (𝑈𝑋 → ((𝜑𝑦𝑋) → (𝑈𝐻𝑦) = (𝑦𝐻𝑈)))
10496, 103mpcom 38 . . . 4 ((𝜑𝑦𝑋) → (𝑈𝐻𝑦) = (𝑦𝐻𝑈))
105 iscringd.7 . . . 4 ((𝜑𝑦𝑋) → (𝑦𝐻𝑈) = 𝑦)
106104, 105eqtrd 2840 . . 3 ((𝜑𝑦𝑋) → (𝑈𝐻𝑦) = 𝑦)
1071, 2, 3, 4, 5, 94, 95, 106, 105isrngod 34006 . 2 (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)
1082eleq2d 2871 . . . . . . 7 (𝜑 → (𝑥𝑋𝑥 ∈ ran 𝐺))
1092eleq2d 2871 . . . . . . 7 (𝜑 → (𝑦𝑋𝑦 ∈ ran 𝐺))
110108, 109anbi12d 618 . . . . . 6 (𝜑 → ((𝑥𝑋𝑦𝑋) ↔ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)))
111110biimpar 465 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (𝑥𝑋𝑦𝑋))
112111, 67syldan 581 . . . 4 ((𝜑 ∧ (𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))
113112ralrimivva 3159 . . 3 (𝜑 → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥𝐻𝑦) = (𝑦𝐻𝑥))
114 rnexg 7324 . . . . . . . 8 (𝐺 ∈ AbelOp → ran 𝐺 ∈ V)
1151, 114syl 17 . . . . . . 7 (𝜑 → ran 𝐺 ∈ V)
1162, 115eqeltrd 2885 . . . . . 6 (𝜑𝑋 ∈ V)
117 xpexg 7186 . . . . . 6 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
118116, 116, 117syl2anc 575 . . . . 5 (𝜑 → (𝑋 × 𝑋) ∈ V)
119 fex 6710 . . . . 5 ((𝐻:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐻 ∈ V)
1203, 118, 119syl2anc 575 . . . 4 (𝜑𝐻 ∈ V)
121 iscom2 34103 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐻 ∈ V) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
1221, 120, 121syl2anc 575 . . 3 (𝜑 → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥𝐻𝑦) = (𝑦𝐻𝑥)))
123113, 122mpbird 248 . 2 (𝜑 → ⟨𝐺, 𝐻⟩ ∈ Com2)
124 iscrngo 34104 . 2 (⟨𝐺, 𝐻⟩ ∈ CRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ ⟨𝐺, 𝐻⟩ ∈ Com2))
125107, 123, 124sylanbrc 574 1 (𝜑 → ⟨𝐺, 𝐻⟩ ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wral 3096  Vcvv 3391  cop 4376   × cxp 5309  ran crn 5312  wf 6093  (class class class)co 6870  GrpOpcgr 27671  AbelOpcablo 27726  RingOpscrngo 34002  Com2ccm2 34097  CRingOpsccring 34101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-grpo 27675  df-ablo 27727  df-rngo 34003  df-com2 34098  df-crngo 34102
This theorem is referenced by: (None)
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