Theorem omeu 8623

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 8620	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
2		opex 5469	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
3	2	isseti 3498	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩
4		19.41v 1949	. . . . . . . 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 709	. . . . . . 7 ⊢ (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
6	5	rexbii 3094	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
7		rexcom4 3288	. . . . . 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 3094	. . . 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 3288	. . . 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 218	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
13		simp2rl 1243	. . . . . . . . . . 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 1247	. . . . . . . . . . . 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 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 𝑥) +o 𝑦) = 𝐵)
16		simp3rr 1248	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . 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 1204	. . . . . . . . . . . . . . 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 1206	. . . . . . . . . . . . . . 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 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))) → 𝑥 ∈ On)
21		simp2lr 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))) → 𝑦 ∈ 𝐴)
22		simp3ll 1245	. . . . . . . . . . . . . . 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 1246	. . . . . . . . . . . . . . 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 8622	. . . . . . . . . . . . . . 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 1388	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 4875	. . . . . . . . . . . . 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 2780	. . . . . . . . . . 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 2780	. . . . . . . . . 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 1122	. . . . . . . . 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 3212	. . . . . 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 3212	. . . 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 1928	. 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 4873	. . . . . . 7 ⊢ (𝑥 = 𝑟 → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑦⟩)
40	39	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = 𝑟 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑦⟩))
41		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑟 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑟))
42	41	oveq1d 7446	. . . . . . 7 ⊢ (𝑥 = 𝑟 → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑦))
43	42	eqeq1d 2739	. . . . . 6 ⊢ (𝑥 = 𝑟 → (((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵))
44	40, 43	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑟 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵)))
45		opeq2 4874	. . . . . . 7 ⊢ (𝑦 = 𝑠 → ⟨𝑟, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
46	45	eqeq2d 2748	. . . . . 6 ⊢ (𝑦 = 𝑠 → (𝑧 = ⟨𝑟, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑠⟩))
47		oveq2 7439	. . . . . . 7 ⊢ (𝑦 = 𝑠 → ((𝐴 ·o 𝑟) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠))
48	47	eqeq1d 2739	. . . . . 6 ⊢ (𝑦 = 𝑠 → (((𝐴 ·o 𝑟) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))
49	46, 48	anbi12d 632	. . . . 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 3242	. . . 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 2741	. . . . . 6 ⊢ (𝑧 = 𝑡 → (𝑧 = ⟨𝑟, 𝑠⟩ ↔ 𝑡 = ⟨𝑟, 𝑠⟩))
52	51	anbi1d 631	. . . . 5 ⊢ (𝑧 = 𝑡 → ((𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) ↔ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
53	52	2rexbidv 3222	. . . 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 2615	. 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 583	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 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2568   ≠ wne 2940  ∃wrex 3070  ∅c0 4333  ⟨cop 4632  Oncon0 6384  (class class class)co 7431   +_o coa 8503   ·_o comu 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511

This theorem is referenced by:  oeeui  8640  omxpenlem  9113  onexomgt  43253