Theorem omeu 8629

Description: The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
omeu	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧

Proof of Theorem omeu

Dummy variables 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omeulem1 8626	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
2		opex 5437	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
3	2	isseti 3472	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩
4		19.41v 1984	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
5	3, 4	mpbiran 722	. . . . . . 7 ⊢ (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
6	5	rexbii 3113	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
7		rexcom4 3293	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑧∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
8	6, 7	bitr3i 280	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑧∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
9	8	rexbii 3113	. . . 4 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑥 ∈ On ∃𝑧∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
10		rexcom4 3293	. . . 4 ⊢ (∃𝑥 ∈ On ∃𝑧∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
11	9, 10	bitri 278	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
12	1, 11	sylib 221	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
13		simp2rl 1261	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑧 = ⟨𝑥, 𝑦⟩)
14		simp3rl 1265	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑡 = ⟨𝑟, 𝑠⟩)
15		simp2rr 1262	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
16		simp3rr 1266	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)
17	15, 16	eqtr4d 2801	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠))
18		simp11 1222	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝐴 ∈ On)
19		simp13 1224	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝐴 ≠ ∅)
20		simp2ll 1259	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑥 ∈ On)
21		simp2lr 1260	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑦 ∈ 𝐴)
22		simp3ll 1263	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑟 ∈ On)
23		simp3lr 1264	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑠 ∈ 𝐴)
24		omopth2 8628	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑟 ∈ On ∧ 𝑠 ∈ 𝐴)) → (((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠) ↔ (𝑥 = 𝑟 ∧ 𝑦 = 𝑠)))
25	18, 19, 20, 21, 22, 23, 24	syl222anc 1414	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → (((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠) ↔ (𝑥 = 𝑟 ∧ 𝑦 = 𝑠)))
26	17, 25	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → (𝑥 = 𝑟 ∧ 𝑦 = 𝑠))
27		opeq12 4845	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
28	26, 27	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
29	14, 28	eqtr4d 2801	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑡 = ⟨𝑥, 𝑦⟩)
30	13, 29	eqtr4d 2801	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑧 = 𝑡)
31	30	3expia 1139	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))) → (((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡))
32	31	exp4b 436	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) → ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡))))
33	32	expd 421	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) → ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡)))))
34	33	rexlimdvv 3222	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) → ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡))))
35	34	imp 412	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) → ((𝑟 ∈ On ∧ 𝑠 ∈ 𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡)))
36	35	rexlimdvv 3222	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) → (∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡))
37	36	expimpd 459	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ((∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡))
38	37	alrimivv 1963	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∀𝑧∀𝑡((∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡))
39		opeq1 4843	. . . . . . 7 ⊢ (𝑥 = 𝑟 → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑦⟩)
40	39	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = 𝑟 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑦⟩))
41		oveq2 7457	. . . . . . . 8 ⊢ (𝑥 = 𝑟 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑟))
42	41	oveq1d 7464	. . . . . . 7 ⊢ (𝑥 = 𝑟 → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑦))
43	42	eqeq1d 2765	. . . . . 6 ⊢ (𝑥 = 𝑟 → (((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵))
44	40, 43	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑟 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵)))
45		opeq2 4844	. . . . . . 7 ⊢ (𝑦 = 𝑠 → ⟨𝑟, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
46	45	eqeq2d 2774	. . . . . 6 ⊢ (𝑦 = 𝑠 → (𝑧 = ⟨𝑟, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑠⟩))
47		oveq2 7457	. . . . . . 7 ⊢ (𝑦 = 𝑠 → ((𝐴 ·o 𝑟) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠))
48	47	eqeq1d 2765	. . . . . 6 ⊢ (𝑦 = 𝑠 → (((𝐴 ·o 𝑟) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))
49	46, 48	anbi12d 644	. . . . 5 ⊢ (𝑦 = 𝑠 → ((𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
50	44, 49	cbvrex2vw 3249	. . . 4 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))
51		eqeq1 2767	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 = ⟨𝑟, 𝑠⟩ ↔ 𝑡 = ⟨𝑟, 𝑠⟩))
52	51	anbi1d 643	. . . . 5 ⊢ (𝑧 = 𝑡 → ((𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) ↔ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
53	52	2rexbidv 3231	. . . 4 ⊢ (𝑧 = 𝑡 → (∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) ↔ ∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
54	50, 53	bitrid 286	. . 3 ⊢ (𝑧 = 𝑡 → (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
55	54	eu4 2643	. 2 ⊢ (∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (∃𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∀𝑧∀𝑡((∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∃𝑟 ∈ On ∃𝑠 ∈ 𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡)))
56	12, 38, 55	sylanbrc 595	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596   ≠ wne 2959  ∃wrex 3090  ∅c0 4281  ⟨cop 4838  Oncon0 6386  (class class class)co 7449   +_o coa 8509   ·_o comu 8510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-oadd 8516  df-omul 8517

This theorem is used by:  oeeui  8647  omxpenlem  9133  omeueq  35830  onexomgt  44498