Theorem oeeu 8599

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 2724	. . . . 5 ⊢ ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} = ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}
2	1	oeeulem 8597	. . . 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 1139	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} ∈ On)
4		fvexd 6897	. . 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 6897	. . 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 2724	. . . 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 2724	. . . 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 2724	. . . 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 8598	. . 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 5504	. 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 1086	. . . . . . . . . . 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 623	. . . . . . . . 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 622	. . . . . . . 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 642	. . . . . . . 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 1842	. . . . . 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 3063	. . . . . 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 3182	. . . . . 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 1843	. . . 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 3187	. . . 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 2571	. 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 217	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 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃!weu 2554  ∃wrex 3062  {crab 3424  Vcvv 3466   ∖ cdif 3938   ⊆ wss 3941  ⟨cop 4627  ⟨cotp 4629  ∪ cuni 4900  ∩ cint 4941  Oncon0 6355  suc csuc 6357  ℩cio 6484  ‘cfv 6534  (class class class)co 7402  1^st c1st 7967  2^nd c2nd 7968  1_oc1o 8455  2_oc2o 8456   +_o coa 8459   ·_o comu 8460   ↑_o coe 8461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468

This theorem is referenced by:  onexoegt  42507  omabs2  42596