Theorem oeeu 8641

Description: The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)

Assertion

Ref	Expression
oeeu	⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 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.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} = ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}
2	1	oeeulem 8639	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} ∈ On ∧ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑o suc ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})))
3	2	simp1d 1143	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} ∈ On)
4		fvexd 6921	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) ∈ V)
5		fvexd 6921	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) ∈ V)
6		eqid 2737	. . . 4 ⊢ (℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵)) = (℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))
7		eqid 2737	. . . 4 ⊢ (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) = (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵)))
8		eqid 2737	. . . 4 ⊢ (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) = (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵)))
9	1, 6, 7, 8	oeeui 8640	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵) ↔ (𝑥 = ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} ∧ 𝑦 = (1st ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))) ∧ 𝑧 = (2nd ‘(℩𝑑∃𝑏 ∈ On ∃𝑐 ∈ (𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴 ↑o ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}) ·o 𝑏) +o 𝑐) = 𝐵))))))
10	3, 4, 5, 9	euotd 5518	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
11		df-3an 1089	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)))
12	11	biancomi 462	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ (𝐴 ↑o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))))
13	12	anbi1i 624	. . . . . . . . 9 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵) ↔ ((𝑧 ∈ (𝐴 ↑o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
14	13	anbi2i 623	. . . . . . . 8 ⊢ ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴 ↑o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
15		an12 645	. . . . . . . 8 ⊢ ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴 ↑o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ((𝑧 ∈ (𝐴 ↑o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
16		anass 468	. . . . . . . 8 ⊢ (((𝑧 ∈ (𝐴 ↑o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴 ↑o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
17	14, 15, 16	3bitri 297	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴 ↑o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
18	17	exbii 1848	. . . . . 6 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴 ↑o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
19		df-rex 3071	. . . . . 6 ⊢ (∃𝑧 ∈ (𝐴 ↑o 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴 ↑o 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))))
20		r19.42v 3191	. . . . . 6 ⊢ (∃𝑧 ∈ (𝐴 ↑o 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
21	18, 19, 20	3bitr2i 299	. . . . 5 ⊢ (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
22	21	2exbii 1849	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑥∃𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
23		r2ex 3196	. . . 4 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵) ↔ ∃𝑥∃𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ ∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)))
24	22, 23	bitr4i 278	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
25	24	eubii 2585	. 2 ⊢ (∃!𝑤∃𝑥∃𝑦∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) ↔ ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))
26	10, 25	sylib 218	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2568  ∃wrex 3070  {crab 3436  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ⟨cop 4632  ⟨cotp 4634  ∪ cuni 4907  ∩ cint 4946  Oncon0 6384  suc csuc 6386  ℩cio 6512  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  1_oc1o 8499  2_oc2o 8500   +_o coa 8503   ·_o comu 8504   ↑_o coe 8505

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-ot 4635  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-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512

This theorem is referenced by:  onexoegt  43256  omabs2  43345