Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isrngo Structured version   Visualization version   GIF version

Theorem isrngo 34118
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 (𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝐻,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem isrngo
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4810 . . . 4 (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2 relrngo 34117 . . . . 5 Rel RingOps
32brrelex1i 5328 . . . 4 (𝐺RingOps𝐻𝐺 ∈ V)
41, 3sylbir 226 . . 3 (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
54a1i 11 . 2 (𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V))
6 elex 3365 . . . 4 (𝐺 ∈ AbelOp → 𝐺 ∈ V)
76ad2antrr 717 . . 3 (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) → 𝐺 ∈ V)
87a1i 11 . 2 (𝐻𝐴 → (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) → 𝐺 ∈ V))
9 df-rngo 34116 . . . . 5 RingOps = {⟨𝑔, ⟩ ∣ ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦)))}
109eleq2i 2836 . . . 4 (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ⟩ ∣ ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦)))})
11 simpl 474 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → 𝑔 = 𝐺)
1211eleq1d 2829 . . . . . . 7 ((𝑔 = 𝐺 = 𝐻) → (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
13 simpr 477 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → = 𝐻)
1411rneqd 5521 . . . . . . . . . 10 ((𝑔 = 𝐺 = 𝐻) → ran 𝑔 = ran 𝐺)
15 isring.1 . . . . . . . . . 10 𝑋 = ran 𝐺
1614, 15syl6eqr 2817 . . . . . . . . 9 ((𝑔 = 𝐺 = 𝐻) → ran 𝑔 = 𝑋)
1716sqxpeqd 5309 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → (ran 𝑔 × ran 𝑔) = (𝑋 × 𝑋))
1813, 17, 16feq123d 6212 . . . . . . 7 ((𝑔 = 𝐺 = 𝐻) → (:(ran 𝑔 × ran 𝑔)⟶ran 𝑔𝐻:(𝑋 × 𝑋)⟶𝑋))
1912, 18anbi12d 624 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ↔ (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋)))
2013oveqd 6859 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
21 eqidd 2766 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → 𝑧 = 𝑧)
2213, 20, 21oveq123d 6863 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑦)𝑧) = ((𝑥𝐻𝑦)𝐻𝑧))
23 eqidd 2766 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → 𝑥 = 𝑥)
2413oveqd 6859 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑦𝑧) = (𝑦𝐻𝑧))
2513, 23, 24oveq123d 6863 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → (𝑥(𝑦𝑧)) = (𝑥𝐻(𝑦𝐻𝑧)))
2622, 25eqeq12d 2780 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → (((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ↔ ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
2711oveqd 6859 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
2813, 23, 27oveq123d 6863 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → (𝑥(𝑦𝑔𝑧)) = (𝑥𝐻(𝑦𝐺𝑧)))
2913oveqd 6859 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑥𝑧) = (𝑥𝐻𝑧))
3011, 20, 29oveq123d 6863 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑦)𝑔(𝑥𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
3128, 30eqeq12d 2780 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → ((𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ↔ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))))
3211oveqd 6859 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3313, 32, 21oveq123d 6863 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑔𝑦)𝑧) = ((𝑥𝐺𝑦)𝐻𝑧))
3411, 29, 24oveq123d 6863 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑧)𝑔(𝑦𝑧)) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
3533, 34eqeq12d 2780 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → (((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧)) ↔ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
3626, 31, 353anbi123d 1560 . . . . . . . . . 10 ((𝑔 = 𝐺 = 𝐻) → ((((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ↔ (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
3716, 36raleqbidv 3300 . . . . . . . . 9 ((𝑔 = 𝐺 = 𝐻) → (∀𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ↔ ∀𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
3816, 37raleqbidv 3300 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → (∀𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ↔ ∀𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
3916, 38raleqbidv 3300 . . . . . . 7 ((𝑔 = 𝐺 = 𝐻) → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
4020eqeq1d 2767 . . . . . . . . . 10 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑦) = 𝑦 ↔ (𝑥𝐻𝑦) = 𝑦))
4113oveqd 6859 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → (𝑦𝑥) = (𝑦𝐻𝑥))
4241eqeq1d 2767 . . . . . . . . . 10 ((𝑔 = 𝐺 = 𝐻) → ((𝑦𝑥) = 𝑦 ↔ (𝑦𝐻𝑥) = 𝑦))
4340, 42anbi12d 624 . . . . . . . . 9 ((𝑔 = 𝐺 = 𝐻) → (((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦) ↔ ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
4416, 43raleqbidv 3300 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → (∀𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦) ↔ ∀𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
4516, 44rexeqbidv 3301 . . . . . . 7 ((𝑔 = 𝐺 = 𝐻) → (∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦) ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
4639, 45anbi12d 624 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → ((∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦)) ↔ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
4719, 46anbi12d 624 . . . . 5 ((𝑔 = 𝐺 = 𝐻) → (((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦))) ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
4847opelopabga 5149 . . . 4 ((𝐺 ∈ V ∧ 𝐻𝐴) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ⟩ ∣ ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦)))} ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
4910, 48syl5bb 274 . . 3 ((𝐺 ∈ V ∧ 𝐻𝐴) → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
5049expcom 402 . 2 (𝐻𝐴 → (𝐺 ∈ V → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))))
515, 8, 50pm5.21ndd 370 1 (𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cop 4340   class class class wbr 4809  {copab 4871   × cxp 5275  ran crn 5278  wf 6064  (class class class)co 6842  AbelOpcablo 27855  RingOpscrngo 34115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-rngo 34116
This theorem is referenced by:  isrngod  34119  rngoi  34120
  Copyright terms: Public domain W3C validator