Theorem oeeu 8605

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 2761	. . . . 5 ⊢ ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} = ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)}
2	1	oeeulem 8603	. . . 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 1160	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∪ ∩ {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑎)} ∈ On)
4		fvexd 6898	. . 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 6898	. . 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 2761	. . . 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 2761	. . . 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 2761	. . . 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 8604	. . 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 5486	. 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 1105	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o)) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)))
12	11	biancomi 468	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o) ∧ 𝑧 ∈ (𝐴 ↑o 𝑥)) ↔ (𝑧 ∈ (𝐴 ↑o 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1o))))
13	12	anbi1i 636	. . . . . . . . 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 635	. . . . . . . 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 658	. . . . . . . 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 474	. . . . . . . 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 300	. . . . . . 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 1881	. . . . . 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 3088	. . . . . 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 3195	. . . . . 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 302	. . . . 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 1882	. . . 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 3200	. . . 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 281	. . 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 2611	. 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 221	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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2594  ∃wrex 3087  {crab 3413  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ⟨cop 4590  ⟨cotp 4592  ∪ cuni 4867  ∩ cint 4907  Oncon0 6361  suc csuc 6363  ℩cio 6491  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998  1_oc1o 8462  2_oc2o 8463   +_o coa 8466   ·_o comu 8467   ↑_o coe 8468

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-oexp 8475

This theorem is used by:  onexoegt  44230  omabs2  44318