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Theorem isrngo 36760
Description: The predicate "is a (unital) ring." Definition of "ring with unit" in [Schechter] p. 187. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isring.1 𝑋 = ran 𝐺
Assertion
Ref Expression
isrngo (𝐻 ∈ 𝐴 β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐺   π‘₯,𝐻,𝑦,𝑧   π‘₯,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑧)
Proof of Theorem isrngo
Dummy variables 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5149 . . . 4 (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2 relrngo 36759 . . . . 5 Rel RingOps
32brrelex1i 5732 . . . 4 (𝐺RingOps𝐻 β†’ 𝐺 ∈ V)
41, 3sylbir 234 . . 3 (⟨𝐺, 𝐻⟩ ∈ RingOps β†’ 𝐺 ∈ V)
54a1i 11 . 2 (𝐻 ∈ 𝐴 β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps β†’ 𝐺 ∈ V))
6 elex 3492 . . . 4 (𝐺 ∈ AbelOp β†’ 𝐺 ∈ V)
76ad2antrr 724 . . 3 (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))) β†’ 𝐺 ∈ V)
87a1i 11 . 2 (𝐻 ∈ 𝐴 β†’ (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))) β†’ 𝐺 ∈ V))
9 df-rngo 36758 . . . . 5 RingOps = {βŸ¨π‘”, β„ŽβŸ© ∣ ((𝑔 ∈ AbelOp ∧ β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔) ∧ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ∧ βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦)))}
109eleq2i 2825 . . . 4 (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {βŸ¨π‘”, β„ŽβŸ© ∣ ((𝑔 ∈ AbelOp ∧ β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔) ∧ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ∧ βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦)))})
11 simpl 483 . . . . . . . 8 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ 𝑔 = 𝐺)
1211eleq1d 2818 . . . . . . 7 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
13 simpr 485 . . . . . . . 8 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ β„Ž = 𝐻)
1411rneqd 5937 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ran 𝑔 = ran 𝐺)
15 isring.1 . . . . . . . . . 10 𝑋 = ran 𝐺
1614, 15eqtr4di 2790 . . . . . . . . 9 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ran 𝑔 = 𝑋)
1716sqxpeqd 5708 . . . . . . . 8 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (ran 𝑔 Γ— ran 𝑔) = (𝑋 Γ— 𝑋))
1813, 17, 16feq123d 6706 . . . . . . 7 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹))
1912, 18anbi12d 631 . . . . . 6 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((𝑔 ∈ AbelOp ∧ β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔) ↔ (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹)))
2013oveqd 7425 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (π‘₯β„Žπ‘¦) = (π‘₯𝐻𝑦))
21 eqidd 2733 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ 𝑧 = 𝑧)
2213, 20, 21oveq123d 7429 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((π‘₯β„Žπ‘¦)β„Žπ‘§) = ((π‘₯𝐻𝑦)𝐻𝑧))
23 eqidd 2733 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ π‘₯ = π‘₯)
2413oveqd 7425 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (π‘¦β„Žπ‘§) = (𝑦𝐻𝑧))
2513, 23, 24oveq123d 7429 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (π‘₯β„Ž(π‘¦β„Žπ‘§)) = (π‘₯𝐻(𝑦𝐻𝑧)))
2622, 25eqeq12d 2748 . . . . . . . . . . 11 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ↔ ((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧))))
2711oveqd 7425 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
2813, 23, 27oveq123d 7429 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (π‘₯β„Ž(𝑦𝑔𝑧)) = (π‘₯𝐻(𝑦𝐺𝑧)))
2913oveqd 7425 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (π‘₯β„Žπ‘§) = (π‘₯𝐻𝑧))
3011, 20, 29oveq123d 7429 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)))
3128, 30eqeq12d 2748 . . . . . . . . . . 11 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ↔ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧))))
3211oveqd 7425 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (π‘₯𝑔𝑦) = (π‘₯𝐺𝑦))
3313, 32, 21oveq123d 7429 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯𝐺𝑦)𝐻𝑧))
3411, 29, 24oveq123d 7429 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§)) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))
3533, 34eqeq12d 2748 . . . . . . . . . . 11 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§)) ↔ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))))
3626, 31, 353anbi123d 1436 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ↔ (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
3716, 36raleqbidv 3342 . . . . . . . . 9 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ↔ βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
3816, 37raleqbidv 3342 . . . . . . . 8 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ↔ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
3916, 38raleqbidv 3342 . . . . . . 7 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
4020eqeq1d 2734 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((π‘₯β„Žπ‘¦) = 𝑦 ↔ (π‘₯𝐻𝑦) = 𝑦))
4113oveqd 7425 . . . . . . . . . . 11 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (π‘¦β„Žπ‘₯) = (𝑦𝐻π‘₯))
4241eqeq1d 2734 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((π‘¦β„Žπ‘₯) = 𝑦 ↔ (𝑦𝐻π‘₯) = 𝑦))
4340, 42anbi12d 631 . . . . . . . . 9 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦) ↔ ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
4416, 43raleqbidv 3342 . . . . . . . 8 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦) ↔ βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
4516, 44rexeqbidv 3343 . . . . . . 7 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦) ↔ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
4639, 45anbi12d 631 . . . . . 6 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ ((βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ∧ βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦)) ↔ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
4719, 46anbi12d 631 . . . . 5 ((𝑔 = 𝐺 ∧ β„Ž = 𝐻) β†’ (((𝑔 ∈ AbelOp ∧ β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔) ∧ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ∧ βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦))) ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
4847opelopabga 5533 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) β†’ (⟨𝐺, 𝐻⟩ ∈ {βŸ¨π‘”, β„ŽβŸ© ∣ ((𝑔 ∈ AbelOp ∧ β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔) ∧ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ∧ βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦)))} ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
4910, 48bitrid 282 . . 3 ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
5049expcom 414 . 2 (𝐻 ∈ 𝐴 β†’ (𝐺 ∈ V β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))))
515, 8, 50pm5.21ndd 380 1 (𝐻 ∈ 𝐴 β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹) ∧ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474  βŸ¨cop 4634   class class class wbr 5148  {copab 5210   Γ— cxp 5674  ran crn 5677  βŸΆwf 6539  (class class class)co 7408  AbelOpcablo 29792  RingOpscrngo 36757
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-rngo 36758
This theorem is referenced by:  isrngod  36761  rngoi  36762
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