Theorem omabs 7727

Description: Ordinal multiplication is also absorbed by powers of ω. (Contributed by Mario Carneiro, 30-May-2015.)

Assertion

Ref	Expression
omabs	⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))

Proof of Theorem omabs

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (ω ↑𝑜 𝑥) = (ω ↑𝑜 ∅))
3	2	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 ∅)))
4	3, 2	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅)))
5	1, 4	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = ∅ → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅))))
6		eleq2 2690	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
7		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦))
8	7	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝑦)))
9	8, 7	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)))
10	6, 9	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))))
11		eleq2 2690	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
12		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 suc 𝑦))
13	12	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)))
14	13, 12	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
15	11, 14	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
16		eleq2 2690	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐵))
17		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝐵))
18	17	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝐵)))
19	18, 17	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵)))
20	16, 19	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
21		noel 3919	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ∅
22	21	pm2.21i 116	. . . . . . . 8 ⊢ (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅))
23	22	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅)))
24		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ω ∈ On)
25		simpll 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ ω)
26		simplr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ∅ ∈ 𝐴)
27		omabslem 7726	. . . . . . . . . . . . . . . 16 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω)
28	24, 25, 26, 27	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 ω) = ω)
29	28	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 ω) = ω)
30		suceq 5790	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
31		df-1o 7560	. . . . . . . . . . . . . . . . . 18 ⊢ 1𝑜 = suc ∅
32	30, 31	syl6eqr 2674	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → suc 𝑦 = 1𝑜)
33	32	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (ω ↑𝑜 suc 𝑦) = (ω ↑𝑜 1𝑜))
34		oe1 7624	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
35	34	ad2antrl 764	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 1𝑜) = ω)
36	33, 35	sylan9eqr 2678	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (ω ↑𝑜 suc 𝑦) = ω)
37	36	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ω))
38	29, 37, 36	3eqtr4d 2666	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))
39	38	ex 450	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
40	39	a1dd 50	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
41		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
42		oesuc 7607	. . . . . . . . . . . . . . . . . 18 ⊢ ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
43	42	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
44	43	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
45		nnon 7071	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
46	45	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ On)
47		oecl 7617	. . . . . . . . . . . . . . . . . 18 ⊢ ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 𝑦) ∈ On)
48	47	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 𝑦) ∈ On)
49		omass 7660	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ (ω ↑𝑜 𝑦) ∈ On ∧ ω ∈ On) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
50	46, 48, 24, 49	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
51	44, 50	eqtr4d 2659	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω))
52	51, 43	eqeq12d 2637	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦) ↔ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = ((ω ↑𝑜 𝑦) ·𝑜 ω)))
53	41, 52	syl5ibr 236	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
54	53	imim2d 57	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
55	54	com23 86	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (∅ ∈ 𝑦 → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
56		simprr 796	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝑦 ∈ On)
57		on0eqel 5845	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
59	40, 55, 58	mpjaod 396	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
60	59	a1dd 50	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
61	60	anassrs 680	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ 𝑦 ∈ On) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
62	61	expcom 451	. . . . . . 7 ⊢ (𝑦 ∈ On → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))))
63	45	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝐴 ∈ On)
64		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ On)
65		simprr 796	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → Lim 𝑥)
66		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
67	65, 66	jctil 560	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (𝑥 ∈ V ∧ Lim 𝑥))
68		limelon 5788	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
69	67, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝑥 ∈ On)
70		oecl 7617	. . . . . . . . . . . . . . . 16 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ↑𝑜 𝑥) ∈ On)
71	64, 69, 70	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) ∈ On)
72	71	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ∈ On)
73		1onn 7719	. . . . . . . . . . . . . . . . . 18 ⊢ 1𝑜 ∈ ω
74	73	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 1𝑜 ∈ ω)
75		ondif2 7582	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
76	64, 74, 75	sylanbrc 698	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ (On ∖ 2𝑜))
77	76	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ (On ∖ 2𝑜))
78	67	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑥 ∈ V ∧ Lim 𝑥))
79		oelimcl 7680	. . . . . . . . . . . . . . 15 ⊢ ((ω ∈ (On ∖ 2𝑜) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (ω ↑𝑜 𝑥))
80	77, 78, 79	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → Lim (ω ↑𝑜 𝑥))
81		omlim 7613	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ ((ω ↑𝑜 𝑥) ∈ On ∧ Lim (ω ↑𝑜 𝑥))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = ∪ 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧))
82	63, 72, 80, 81	syl12anc 1324	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = ∪ 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧))
83		simplrl 800	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ On)
84		oelim2 7675	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ω ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) = ∪ 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦))
85	83, 78, 84	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) = ∪ 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦))
86	85	eleq2d 2687	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ 𝑧 ∈ ∪ 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦)))
87		eliun 4524	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦))
88	86, 87	syl6bb 276	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
89	69	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝑥 ∈ On)
90		anass 681	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ (𝑦 ∈ 𝑥 ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
91		onelon 5748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
92		on0eln0 5780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ On → (∅ ∈ 𝑦 ↔ 𝑦 ≠ ∅))
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (∅ ∈ 𝑦 ↔ 𝑦 ≠ ∅))
94	93	pm5.32da 673	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ≠ ∅)))
95		dif1o 7580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝑥 ∖ 1𝑜) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ≠ ∅))
96	94, 95	syl6bbr 278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ↔ 𝑦 ∈ (𝑥 ∖ 1𝑜)))
97	96	anbi1d 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ (𝑦 ∈ (𝑥 ∖ 1𝑜) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
98	90, 97	syl5bbr 274	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) ↔ (𝑦 ∈ (𝑥 ∖ 1𝑜) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
99	98	rexbidv2 3048	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → (∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
100	89, 99	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
101	88, 100	bitr4d 271	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ ∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
102		r19.29 3072	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ∃𝑦 ∈ 𝑥 ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
103		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)))
104	103	imp 445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∅ ∈ 𝑦) → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))
105	104	anim1i 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)))
106	105	anasss 679	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)))
107	71	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ∈ On)
108		eloni 5733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ω ↑𝑜 𝑥) ∈ On → Ord (ω ↑𝑜 𝑥))
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → Ord (ω ↑𝑜 𝑥))
110		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑧 ∈ (ω ↑𝑜 𝑦))
111	64	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ω ∈ On)
112	69	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑥 ∈ On)
113		simplr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑦 ∈ 𝑥)
114	112, 113, 91	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑦 ∈ On)
115	111, 114, 47	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑦) ∈ On)
116		onelon 5748	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((ω ↑𝑜 𝑦) ∈ On ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → 𝑧 ∈ On)
117	115, 110, 116	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑧 ∈ On)
118	45	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝐴 ∈ On)
119	118	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝐴 ∈ On)
120		simplr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ∅ ∈ 𝐴)
121	120	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
122		omord2 7647	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑧 ∈ On ∧ (ω ↑𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ω ↑𝑜 𝑦) ↔ (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦))))
123	117, 115, 119, 121, 122	syl31anc 1329	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑦) ↔ (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦))))
124	110, 123	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦)))
125		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))
126	124, 125	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦))
127	76	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ω ∈ (On ∖ 2𝑜))
128		oeord 7668	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ ω ∈ (On ∖ 2𝑜)) → (𝑦 ∈ 𝑥 ↔ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)))
129	114, 112, 127, 128	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝑦 ∈ 𝑥 ↔ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)))
130	113, 129	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥))
131		ontr1 5771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ω ↑𝑜 𝑥) ∈ On → (((𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦) ∧ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)))
132	107, 131	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (((𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦) ∧ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)))
133	126, 130, 132	mp2and 715	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥))
134		ordelss 5739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord (ω ↑𝑜 𝑥) ∧ (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
135	109, 133, 134	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
136	135	ex 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) → (((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
137	106, 136	syl5 34	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) → (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
138	137	rexlimdva 3031	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∃𝑦 ∈ 𝑥 ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
139	102, 138	syl5 34	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ((∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
140	139	expdimp 453	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
141	101, 140	sylbid 230	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
142	141	ralrimiv 2965	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∀𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
143		iunss 4561	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥) ↔ ∀𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
144	142, 143	sylibr 224	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∪ 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
145	82, 144	eqsstrd 3639	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) ⊆ (ω ↑𝑜 𝑥))
146		simpllr 799	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
147		omword2 7654	. . . . . . . . . . . . 13 ⊢ ((((ω ↑𝑜 𝑥) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (ω ↑𝑜 𝑥) ⊆ (𝐴 ·𝑜 (ω ↑𝑜 𝑥)))
148	72, 63, 146, 147	syl21anc 1325	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ⊆ (𝐴 ·𝑜 (ω ↑𝑜 𝑥)))
149	145, 148	eqssd 3620	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥))
150	149	ex 450	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))
151	150	anassrs 680	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))
152	151	a1dd 50	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥))))
153	152	expcom 451	. . . . . . 7 ⊢ (Lim 𝑥 → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))))
154	5, 10, 15, 20, 23, 62, 153	tfinds3 7064	. . . . . 6 ⊢ (𝐵 ∈ On → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
155	154	com12 32	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
156	155	adantrr 753	. . . 4 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
157	156	imp32 449	. . 3 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
158	157	an32s 846	. 2 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
159		nnm0 7685	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
160	159	ad3antrrr 766	. . 3 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 ∅) = ∅)
161		fnoe 7590	. . . . . . 7 ⊢ ↑𝑜 Fn (On × On)
162		fndm 5990	. . . . . . 7 ⊢ ( ↑𝑜 Fn (On × On) → dom ↑𝑜 = (On × On))
163	161, 162	ax-mp 5	. . . . . 6 ⊢ dom ↑𝑜 = (On × On)
164	163	ndmov 6818	. . . . 5 ⊢ (¬ (ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑𝑜 𝐵) = ∅)
165	164	adantl 482	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (ω ↑𝑜 𝐵) = ∅)
166	165	oveq2d 6666	. . 3 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (𝐴 ·𝑜 ∅))
167	160, 166, 165	3eqtr4d 2666	. 2 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
168	158, 167	pm2.61dan 832	1 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520   × cxp 5112  dom cdm 5114  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725   Fn wfn 5883  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  cnfcom3  8601