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Theorem oeeu 7728
Description: The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
Assertion
Ref Expression
oeeu ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧

Proof of Theorem oeeu
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . . . 6 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}
21oeeulem 7726 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On ∧ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})))
32simp1d 1093 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On)
4 elex 3243 . . . 4 ( {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ V)
53, 4syl 17 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ V)
6 fvexd 6241 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
7 fvexd 6241 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
8 eqid 2651 . . . 4 (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)) = (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))
9 eqid 2651 . . . 4 (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
10 eqid 2651 . . . 4 (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
111, 8, 9, 10oeeui 7727 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ (𝑥 = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∧ 𝑦 = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∧ 𝑧 = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))))))
125, 6, 7, 11euotd 5004 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
13 df-3an 1056 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)))
14 ancom 465 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
1513, 14bitri 264 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
1615anbi1i 731 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
1716anbi2i 730 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
18 an12 855 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
19 anass 682 . . . . . . . 8 (((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
2017, 18, 193bitri 286 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
2120exbii 1814 . . . . . 6 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
22 df-rex 2947 . . . . . 6 (∃𝑧 ∈ (𝐴𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
23 r19.42v 3121 . . . . . 6 (∃𝑧 ∈ (𝐴𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
2421, 22, 233bitr2i 288 . . . . 5 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
25242exbii 1815 . . . 4 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
26 r2ex 3090 . . . 4 (∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
2725, 26bitr4i 267 . . 3 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
2827eubii 2520 . 2 (∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
2912, 28sylib 208 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  cop 4216  cotp 4218   cuni 4468   cint 4507  Oncon0 5761  suc csuc 5763  cio 5887  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  1𝑜c1o 7598  2𝑜c2o 7599   +𝑜 coa 7602   ·𝑜 comu 7603  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by: (None)
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