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Theorem omeu 8462
Description: The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
omeu ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧𝑥 ∈ On ∃𝑦𝐴 (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.
StepHypRef Expression
1 omeulem1 8459 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
2 opex 5396 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
32isseti 3456 . . . . . . . 8 𝑧 𝑧 = ⟨𝑥, 𝑦
4 19.41v 1952 . . . . . . . 8 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
53, 4mpbiran 706 . . . . . . 7 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
65rexbii 3094 . . . . . 6 (∃𝑦𝐴𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
7 rexcom4 3268 . . . . . 6 (∃𝑦𝐴𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑧𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
86, 7bitr3i 276 . . . . 5 (∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑧𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
98rexbii 3094 . . . 4 (∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑥 ∈ On ∃𝑧𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
10 rexcom4 3268 . . . 4 (∃𝑥 ∈ On ∃𝑧𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
119, 10bitri 274 . . 3 (∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
121, 11sylib 217 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
13 simp2rl 1241 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑧 = ⟨𝑥, 𝑦⟩)
14 simp3rl 1245 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑡 = ⟨𝑟, 𝑠⟩)
15 simp2rr 1242 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
16 simp3rr 1246 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)
1715, 16eqtr4d 2780 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠))
18 simp11 1202 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝐴 ∈ On)
19 simp13 1204 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝐴 ≠ ∅)
20 simp2ll 1239 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑥 ∈ On)
21 simp2lr 1240 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑦𝐴)
22 simp3ll 1243 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑟 ∈ On)
23 simp3lr 1244 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑠𝐴)
24 omopth2 8461 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑟 ∈ On ∧ 𝑠𝐴)) → (((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠) ↔ (𝑥 = 𝑟𝑦 = 𝑠)))
2518, 19, 20, 21, 22, 23, 24syl222anc 1385 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → (((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠) ↔ (𝑥 = 𝑟𝑦 = 𝑠)))
2617, 25mpbid 231 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → (𝑥 = 𝑟𝑦 = 𝑠))
27 opeq12 4815 . . . . . . . . . . . . 13 ((𝑥 = 𝑟𝑦 = 𝑠) → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
2826, 27syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
2914, 28eqtr4d 2780 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑡 = ⟨𝑥, 𝑦⟩)
3013, 29eqtr4d 2780 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) ∧ ((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))) → 𝑧 = 𝑡)
31303expia 1120 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))) → (((𝑟 ∈ On ∧ 𝑠𝐴) ∧ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡))
3231exp4b 431 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) → ((𝑟 ∈ On ∧ 𝑠𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡))))
3332expd 416 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦𝐴) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) → ((𝑟 ∈ On ∧ 𝑠𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡)))))
3433rexlimdvv 3201 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) → ((𝑟 ∈ On ∧ 𝑠𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡))))
3534imp 407 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) → ((𝑟 ∈ On ∧ 𝑠𝐴) → ((𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡)))
3635rexlimdvv 3201 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) → (∃𝑟 ∈ On ∃𝑠𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) → 𝑧 = 𝑡))
3736expimpd 454 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ((∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∃𝑟 ∈ On ∃𝑠𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡))
3837alrimivv 1930 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∀𝑧𝑡((∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∃𝑟 ∈ On ∃𝑠𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡))
39 opeq1 4813 . . . . . . 7 (𝑥 = 𝑟 → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑦⟩)
4039eqeq2d 2748 . . . . . 6 (𝑥 = 𝑟 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑦⟩))
41 oveq2 7321 . . . . . . . 8 (𝑥 = 𝑟 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑟))
4241oveq1d 7328 . . . . . . 7 (𝑥 = 𝑟 → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑦))
4342eqeq1d 2739 . . . . . 6 (𝑥 = 𝑟 → (((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵))
4440, 43anbi12d 631 . . . . 5 (𝑥 = 𝑟 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵)))
45 opeq2 4814 . . . . . . 7 (𝑦 = 𝑠 → ⟨𝑟, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
4645eqeq2d 2748 . . . . . 6 (𝑦 = 𝑠 → (𝑧 = ⟨𝑟, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑠⟩))
47 oveq2 7321 . . . . . . 7 (𝑦 = 𝑠 → ((𝐴 ·o 𝑟) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠))
4847eqeq1d 2739 . . . . . 6 (𝑦 = 𝑠 → (((𝐴 ·o 𝑟) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))
4946, 48anbi12d 631 . . . . 5 (𝑦 = 𝑠 → ((𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
5044, 49cbvrex2vw 3225 . . . 4 (∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑟 ∈ On ∃𝑠𝐴 (𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))
51 eqeq1 2741 . . . . . 6 (𝑧 = 𝑡 → (𝑧 = ⟨𝑟, 𝑠⟩ ↔ 𝑡 = ⟨𝑟, 𝑠⟩))
5251anbi1d 630 . . . . 5 (𝑧 = 𝑡 → ((𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) ↔ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
53522rexbidv 3210 . . . 4 (𝑧 = 𝑡 → (∃𝑟 ∈ On ∃𝑠𝐴 (𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) ↔ ∃𝑟 ∈ On ∃𝑠𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
5450, 53bitrid 282 . . 3 (𝑧 = 𝑡 → (∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑟 ∈ On ∃𝑠𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
5554eu4 2616 . 2 (∃!𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (∃𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∀𝑧𝑡((∃𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ∧ ∃𝑟 ∈ On ∃𝑠𝐴 (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)) → 𝑧 = 𝑡)))
5612, 38, 55sylanbrc 583 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wal 1538   = wceq 1540  wex 1780  wcel 2105  ∃!weu 2567  wne 2941  wrex 3071  c0 4266  cop 4575  Oncon0 6286  (class class class)co 7313   +o coa 8339   ·o comu 8340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-int 4891  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-om 7756  df-2nd 7875  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286  df-1o 8342  df-oadd 8346  df-omul 8347
This theorem is referenced by:  oeeui  8479  omxpenlem  8913
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