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Theorem omeu 8200
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 8197 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
2 opex 5347 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
32isseti 3506 . . . . . . . 8 𝑧 𝑧 = ⟨𝑥, 𝑦
4 19.41v 1941 . . . . . . . 8 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
53, 4mpbiran 705 . . . . . . 7 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
65rexbii 3244 . . . . . 6 (∃𝑦𝐴𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
7 rexcom4 3246 . . . . . 6 (∃𝑦𝐴𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ ∃𝑧𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
86, 7bitr3i 278 . . . . 5 (∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑧𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
98rexbii 3244 . . . 4 (∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑥 ∈ On ∃𝑧𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
10 rexcom4 3246 . . . 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 276 . . 3 (∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ∃𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
121, 11sylib 219 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵))
13 simp2rl 1234 . . . . . . . . . . 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 1238 . . . . . . . . . . . 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 1235 . . . . . . . . . . . . . . 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 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))) → ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)
1715, 16eqtr4d 2856 . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . . . 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 8199 . . . . . . . . . . . . . . 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 1378 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 4797 . . . . . . . . . . . . 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 2856 . . . . . . . . . . 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 2856 . . . . . . . . . 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 1113 . . . . . . . . 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 3290 . . . . . 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 3290 . . . 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 1920 . 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 4795 . . . . . . 7 (𝑥 = 𝑟 → ⟨𝑥, 𝑦⟩ = ⟨𝑟, 𝑦⟩)
4039eqeq2d 2829 . . . . . 6 (𝑥 = 𝑟 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑦⟩))
41 oveq2 7153 . . . . . . . 8 (𝑥 = 𝑟 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑟))
4241oveq1d 7160 . . . . . . 7 (𝑥 = 𝑟 → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑦))
4342eqeq1d 2820 . . . . . 6 (𝑥 = 𝑟 → (((𝐴 ·o 𝑥) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵))
4440, 43anbi12d 630 . . . . 5 (𝑥 = 𝑟 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵)))
45 opeq2 4796 . . . . . . 7 (𝑦 = 𝑠 → ⟨𝑟, 𝑦⟩ = ⟨𝑟, 𝑠⟩)
4645eqeq2d 2829 . . . . . 6 (𝑦 = 𝑠 → (𝑧 = ⟨𝑟, 𝑦⟩ ↔ 𝑧 = ⟨𝑟, 𝑠⟩))
47 oveq2 7153 . . . . . . 7 (𝑦 = 𝑠 → ((𝐴 ·o 𝑟) +o 𝑦) = ((𝐴 ·o 𝑟) +o 𝑠))
4847eqeq1d 2820 . . . . . 6 (𝑦 = 𝑠 → (((𝐴 ·o 𝑟) +o 𝑦) = 𝐵 ↔ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵))
4946, 48anbi12d 630 . . . . 5 (𝑦 = 𝑠 → ((𝑧 = ⟨𝑟, 𝑦⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑦) = 𝐵) ↔ (𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
5044, 49cbvrex2vw 3460 . . . 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 2822 . . . . . 6 (𝑧 = 𝑡 → (𝑧 = ⟨𝑟, 𝑠⟩ ↔ 𝑡 = ⟨𝑟, 𝑠⟩))
5251anbi1d 629 . . . . 5 (𝑧 = 𝑡 → ((𝑧 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵) ↔ (𝑡 = ⟨𝑟, 𝑠⟩ ∧ ((𝐴 ·o 𝑟) +o 𝑠) = 𝐵)))
53522rexbidv 3297 . . . 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, 53syl5bb 284 . . 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 2692 . 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 207  wa 396  w3a 1079  wal 1526   = wceq 1528  wex 1771  wcel 2105  ∃!weu 2646  wne 3013  wrex 3136  c0 4288  cop 4563  Oncon0 6184  (class class class)co 7145   +o coa 8088   ·o comu 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096
This theorem is referenced by:  oeeui  8217  omxpenlem  8606
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