Theorem omeu 8581

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 8578	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
2		opex 5455	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
3	2	isseti 3482	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩
4		19.41v 1945	. . . . . . . 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 706	. . . . . . 7 ⊢ (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
6	5	rexbii 3086	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
7		rexcom4 3277	. . . . . 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 277	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑧∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
9	8	rexbii 3086	. . . 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 3277	. . . 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 275	. . 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 217	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
13		simp2rl 1239	. . . . . . . . . . 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 1243	. . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . 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 1244	. . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . 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 1237	. . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . 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 1241	. . . . . . . . . . . . . . 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 1242	. . . . . . . . . . . . . . 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 8580	. . . . . . . . . . . . . . 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 1383	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 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 4868	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑟 ∧ 𝑦 = 𝑠) → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
28	26, 27	syl 17	. . . . . . . . . . . 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 2767	. . . . . . . . . . 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 2767	. . . . . . . . . 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 1118	. . . . . . . . 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 430	. . . . . . . 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 415	. . . . . . 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 3202	. . . . . 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 406	. . . . 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 3202	. . . 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 453	. . 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 1923	. 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 4866	. . . . . . 7 ⊢ (𝑥 = 𝑟 → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑦⟩)
40	39	eqeq2d 2735	. . . . . 6 ⊢ (𝑥 = 𝑟 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑦⟩))
41		oveq2 7410	. . . . . . . 8 ⊢ (𝑥 = 𝑟 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑟))
42	41	oveq1d 7417	. . . . . . 7 ⊢ (𝑥 = 𝑟 → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑦))
43	42	eqeq1d 2726	. . . . . 6 ⊢ (𝑥 = 𝑟 → (((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵))
44	40, 43	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝑟 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵)))
45		opeq2 4867	. . . . . . 7 ⊢ (𝑦 = 𝑠 → ⟨𝑟, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
46	45	eqeq2d 2735	. . . . . 6 ⊢ (𝑦 = 𝑠 → (𝑧 = ⟨𝑟, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑠⟩))
47		oveq2 7410	. . . . . . 7 ⊢ (𝑦 = 𝑠 → ((𝐴 ·o 𝑟) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠))
48	47	eqeq1d 2726	. . . . . 6 ⊢ (𝑦 = 𝑠 → (((𝐴 ·o 𝑟) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))
49	46, 48	anbi12d 630	. . . . 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 3231	. . . 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 2728	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 = ⟨𝑟, 𝑠⟩ ↔ 𝑡 = ⟨𝑟, 𝑠⟩))
52	51	anbi1d 629	. . . . 5 ⊢ (𝑧 = 𝑡 → ((𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) ↔ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
53	52	2rexbidv 3211	. . . 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 283	. . 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 2603	. 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 582	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃!weu 2554   ≠ wne 2932  ∃wrex 3062  ∅c0 4315  ⟨cop 4627  Oncon0 6355  (class class class)co 7402   +_o coa 8459   ·_o comu 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467

This theorem is referenced by:  oeeui  8598  omxpenlem  9070  onexomgt  42504