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

Theorem isrngo 35351
 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 5031 . . . 4 (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2 relrngo 35350 . . . . 5 Rel RingOps
32brrelex1i 5572 . . . 4 (𝐺RingOps𝐻𝐺 ∈ V)
41, 3sylbir 238 . . 3 (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
54a1i 11 . 2 (𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V))
6 elex 3459 . . . 4 (𝐺 ∈ AbelOp → 𝐺 ∈ V)
76ad2antrr 725 . . 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 35349 . . . . 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 2881 . . . 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 486 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → 𝑔 = 𝐺)
1211eleq1d 2874 . . . . . . 7 ((𝑔 = 𝐺 = 𝐻) → (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
13 simpr 488 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → = 𝐻)
1411rneqd 5772 . . . . . . . . . 10 ((𝑔 = 𝐺 = 𝐻) → ran 𝑔 = ran 𝐺)
15 isring.1 . . . . . . . . . 10 𝑋 = ran 𝐺
1614, 15eqtr4di 2851 . . . . . . . . 9 ((𝑔 = 𝐺 = 𝐻) → ran 𝑔 = 𝑋)
1716sqxpeqd 5551 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → (ran 𝑔 × ran 𝑔) = (𝑋 × 𝑋))
1813, 17, 16feq123d 6476 . . . . . . 7 ((𝑔 = 𝐺 = 𝐻) → (:(ran 𝑔 × ran 𝑔)⟶ran 𝑔𝐻:(𝑋 × 𝑋)⟶𝑋))
1912, 18anbi12d 633 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ↔ (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋)))
2013oveqd 7152 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
21 eqidd 2799 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → 𝑧 = 𝑧)
2213, 20, 21oveq123d 7156 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑦)𝑧) = ((𝑥𝐻𝑦)𝐻𝑧))
23 eqidd 2799 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → 𝑥 = 𝑥)
2413oveqd 7152 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑦𝑧) = (𝑦𝐻𝑧))
2513, 23, 24oveq123d 7156 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → (𝑥(𝑦𝑧)) = (𝑥𝐻(𝑦𝐻𝑧)))
2622, 25eqeq12d 2814 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → (((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ↔ ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
2711oveqd 7152 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
2813, 23, 27oveq123d 7156 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → (𝑥(𝑦𝑔𝑧)) = (𝑥𝐻(𝑦𝐺𝑧)))
2913oveqd 7152 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑥𝑧) = (𝑥𝐻𝑧))
3011, 20, 29oveq123d 7156 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑦)𝑔(𝑥𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
3128, 30eqeq12d 2814 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → ((𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ↔ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))))
3211oveqd 7152 . . . . . . . . . . . . 13 ((𝑔 = 𝐺 = 𝐻) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3313, 32, 21oveq123d 7156 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑔𝑦)𝑧) = ((𝑥𝐺𝑦)𝐻𝑧))
3411, 29, 24oveq123d 7156 . . . . . . . . . . . 12 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑧)𝑔(𝑦𝑧)) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
3533, 34eqeq12d 2814 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → (((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧)) ↔ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
3626, 31, 353anbi123d 1433 . . . . . . . . . 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 3354 . . . . . . . . 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 3354 . . . . . . . 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 3354 . . . . . . 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 2800 . . . . . . . . . 10 ((𝑔 = 𝐺 = 𝐻) → ((𝑥𝑦) = 𝑦 ↔ (𝑥𝐻𝑦) = 𝑦))
4113oveqd 7152 . . . . . . . . . . 11 ((𝑔 = 𝐺 = 𝐻) → (𝑦𝑥) = (𝑦𝐻𝑥))
4241eqeq1d 2800 . . . . . . . . . 10 ((𝑔 = 𝐺 = 𝐻) → ((𝑦𝑥) = 𝑦 ↔ (𝑦𝐻𝑥) = 𝑦))
4340, 42anbi12d 633 . . . . . . . . 9 ((𝑔 = 𝐺 = 𝐻) → (((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦) ↔ ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
4416, 43raleqbidv 3354 . . . . . . . 8 ((𝑔 = 𝐺 = 𝐻) → (∀𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦) ↔ ∀𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
4516, 44rexeqbidv 3355 . . . . . . 7 ((𝑔 = 𝐺 = 𝐻) → (∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦) ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
4639, 45anbi12d 633 . . . . . 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 633 . . . . 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 5385 . . . 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 286 . . 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 417 . 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 384 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441  ⟨cop 4531   class class class wbr 5030  {copab 5092   × cxp 5517  ran crn 5520  ⟶wf 6320  (class class class)co 7135  AbelOpcablo 28334  RingOpscrngo 35348 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-rngo 35349 This theorem is referenced by:  isrngod  35352  rngoi  35353
 Copyright terms: Public domain W3C validator