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Theorem omabs 7898
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.
StepHypRef Expression
1 eleq2 2828 . . . . . . . 8 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2 oveq2 6822 . . . . . . . . . 10 (𝑥 = ∅ → (ω ↑𝑜 𝑥) = (ω ↑𝑜 ∅))
32oveq2d 6830 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 ∅)))
43, 2eqeq12d 2775 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅)))
51, 4imbi12d 333 . . . . . . 7 (𝑥 = ∅ → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅))))
6 eleq2 2828 . . . . . . . 8 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
7 oveq2 6822 . . . . . . . . . 10 (𝑥 = 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦))
87oveq2d 6830 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝑦)))
98, 7eqeq12d 2775 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)))
106, 9imbi12d 333 . . . . . . 7 (𝑥 = 𝑦 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))))
11 eleq2 2828 . . . . . . . 8 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
12 oveq2 6822 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 suc 𝑦))
1312oveq2d 6830 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)))
1413, 12eqeq12d 2775 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
1511, 14imbi12d 333 . . . . . . 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 2828 . . . . . . . 8 (𝑥 = 𝐵 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐵))
17 oveq2 6822 . . . . . . . . . 10 (𝑥 = 𝐵 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝐵))
1817oveq2d 6830 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝐵)))
1918, 17eqeq12d 2775 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵)))
2016, 19imbi12d 333 . . . . . . 7 (𝑥 = 𝐵 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
21 noel 4062 . . . . . . . . 9 ¬ ∅ ∈ ∅
2221pm2.21i 116 . . . . . . . 8 (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅))
2322a1i 11 . . . . . . 7 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅)))
24 simprl 811 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ω ∈ On)
25 simpll 807 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ ω)
26 simplr 809 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ∅ ∈ 𝐴)
27 omabslem 7897 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω)
2824, 25, 26, 27syl3anc 1477 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 ω) = ω)
2928adantr 472 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 ω) = ω)
30 suceq 5951 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → suc 𝑦 = suc ∅)
31 df-1o 7730 . . . . . . . . . . . . . . . . . 18 1𝑜 = suc ∅
3230, 31syl6eqr 2812 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → suc 𝑦 = 1𝑜)
3332oveq2d 6830 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (ω ↑𝑜 suc 𝑦) = (ω ↑𝑜 1𝑜))
34 oe1 7795 . . . . . . . . . . . . . . . . 17 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
3534ad2antrl 766 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 1𝑜) = ω)
3633, 35sylan9eqr 2816 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (ω ↑𝑜 suc 𝑦) = ω)
3736oveq2d 6830 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ω))
3829, 37, 363eqtr4d 2804 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))
3938ex 449 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
4039a1dd 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 6821 . . . . . . . . . . . . . 14 ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
42 oesuc 7778 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
4342adantl 473 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
4443oveq2d 6830 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
45 nnon 7237 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ω → 𝐴 ∈ On)
4645ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ On)
47 oecl 7788 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 𝑦) ∈ On)
4847adantl 473 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 𝑦) ∈ On)
49 omass 7831 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ (ω ↑𝑜 𝑦) ∈ On ∧ ω ∈ On) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
5046, 48, 24, 49syl3anc 1477 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
5144, 50eqtr4d 2797 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω))
5251, 43eqeq12d 2775 . . . . . . . . . . . . . 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)))
5341, 52syl5ibr 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)))
5453imim2d 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))))
5554com23 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 813 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝑦 ∈ On)
57 on0eqel 6006 . . . . . . . . . . . 12 (𝑦 ∈ On → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
5856, 57syl 17 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
5940, 55, 58mpjaod 395 . . . . . . . . . 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)))
6059a1dd 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))))
6160anassrs 683 . . . . . . . 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))))
6261expcom 450 . . . . . . 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)))))
6345ad3antrrr 768 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝐴 ∈ On)
64 simprl 811 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ On)
65 simprr 813 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → Lim 𝑥)
66 vex 3343 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
6765, 66jctil 561 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (𝑥 ∈ V ∧ Lim 𝑥))
68 limelon 5949 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
6967, 68syl 17 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝑥 ∈ On)
70 oecl 7788 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ↑𝑜 𝑥) ∈ On)
7164, 69, 70syl2anc 696 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) ∈ On)
7271adantr 472 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ∈ On)
73 1onn 7890 . . . . . . . . . . . . . . . . . 18 1𝑜 ∈ ω
7473a1i 11 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 1𝑜 ∈ ω)
75 ondif2 7753 . . . . . . . . . . . . . . . . 17 (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
7664, 74, 75sylanbrc 701 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ (On ∖ 2𝑜))
7776adantr 472 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ (On ∖ 2𝑜))
7867adantr 472 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑥 ∈ V ∧ Lim 𝑥))
79 oelimcl 7851 . . . . . . . . . . . . . . 15 ((ω ∈ (On ∖ 2𝑜) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (ω ↑𝑜 𝑥))
8077, 78, 79syl2anc 696 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → Lim (ω ↑𝑜 𝑥))
81 omlim 7784 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ ((ω ↑𝑜 𝑥) ∈ On ∧ Lim (ω ↑𝑜 𝑥))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧))
8263, 72, 80, 81syl12anc 1475 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧))
83 simplrl 819 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ On)
84 oelim2 7846 . . . . . . . . . . . . . . . . . . . 20 ((ω ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) = 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦))
8583, 78, 84syl2anc 696 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) = 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦))
8685eleq2d 2825 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ 𝑧 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦)))
87 eliun 4676 . . . . . . . . . . . . . . . . . 18 (𝑧 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦))
8886, 87syl6bb 276 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
8969adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝑥 ∈ On)
90 anass 684 . . . . . . . . . . . . . . . . . . . 20 (((𝑦𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ (𝑦𝑥 ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))))
91 onelon 5909 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
92 on0eln0 5941 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (∅ ∈ 𝑦𝑦 ≠ ∅))
9391, 92syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ On ∧ 𝑦𝑥) → (∅ ∈ 𝑦𝑦 ≠ ∅))
9493pm5.32da 676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ (𝑦𝑥𝑦 ≠ ∅)))
95 dif1o 7751 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝑥 ∖ 1𝑜) ↔ (𝑦𝑥𝑦 ≠ ∅))
9694, 95syl6bbr 278 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ 𝑦 ∈ (𝑥 ∖ 1𝑜)))
9796anbi1d 743 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (((𝑦𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ (𝑦 ∈ (𝑥 ∖ 1𝑜) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
9890, 97syl5bbr 274 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → ((𝑦𝑥 ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) ↔ (𝑦 ∈ (𝑥 ∖ 1𝑜) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
9998rexbidv2 3186 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → (∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
10089, 99syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
10188, 100bitr4d 271 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ ∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))))
102 r19.29 3210 . . . . . . . . . . . . . . . . . 18 ((∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → ∃𝑦𝑥 ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (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)))
104103imp 444 . . . . . . . . . . . . . . . . . . . . . 22 (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∅ ∈ 𝑦) → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))
105104anim1i 593 . . . . . . . . . . . . . . . . . . . . 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)))
106105anasss 682 . . . . . . . . . . . . . . . . . . . 20 (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)))
10771ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ∈ On)
108 eloni 5894 . . . . . . . . . . . . . . . . . . . . . . 23 ((ω ↑𝑜 𝑥) ∈ On → Ord (ω ↑𝑜 𝑥))
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → Ord (ω ↑𝑜 𝑥))
110 simprr 813 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑧 ∈ (ω ↑𝑜 𝑦))
11164ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ω ∈ On)
11269ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑥 ∈ On)
113 simplr 809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑦𝑥)
114112, 113, 91syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑦 ∈ On)
115111, 114, 47syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑦) ∈ On)
116 onelon 5909 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((ω ↑𝑜 𝑦) ∈ On ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → 𝑧 ∈ On)
117115, 110, 116syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑧 ∈ On)
11845ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝐴 ∈ On)
119118ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝐴 ∈ On)
120 simplr 809 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ∅ ∈ 𝐴)
121120ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
122 omord2 7818 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ On ∧ (ω ↑𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ω ↑𝑜 𝑦) ↔ (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦))))
123117, 115, 119, 121, 122syl31anc 1480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑦) ↔ (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦))))
124110, 123mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦)))
125 simprl 811 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))
126124, 125eleqtrd 2841 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦))
12776ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ω ∈ (On ∖ 2𝑜))
128 oeord 7839 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ ω ∈ (On ∖ 2𝑜)) → (𝑦𝑥 ↔ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)))
129114, 112, 127, 128syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝑦𝑥 ↔ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)))
130113, 129mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥))
131 ontr1 5932 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ω ↑𝑜 𝑥) ∈ On → (((𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦) ∧ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)))
132107, 131syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((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)))
133126, 130, 132mp2and 717 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥))
134 ordelss 5900 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord (ω ↑𝑜 𝑥) ∧ (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
135109, 133, 134syl2anc 696 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
136135ex 449 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) → (((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
137106, 136syl5 34 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) → (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
138137rexlimdva 3169 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∃𝑦𝑥 ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
139102, 138syl5 34 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ((∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
140139expdimp 452 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
141101, 140sylbid 230 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
142141ralrimiv 3103 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∀𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
143 iunss 4713 . . . . . . . . . . . . . 14 ( 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥) ↔ ∀𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
144142, 143sylibr 224 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
14582, 144eqsstrd 3780 . . . . . . . . . . . 12 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) ⊆ (ω ↑𝑜 𝑥))
146 simpllr 817 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
147 omword2 7825 . . . . . . . . . . . . 13 ((((ω ↑𝑜 𝑥) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (ω ↑𝑜 𝑥) ⊆ (𝐴 ·𝑜 (ω ↑𝑜 𝑥)))
14872, 63, 146, 147syl21anc 1476 . . . . . . . . . . . 12 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ⊆ (𝐴 ·𝑜 (ω ↑𝑜 𝑥)))
149145, 148eqssd 3761 . . . . . . . . . . 11 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥))
150149ex 449 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))
151150anassrs 683 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))
152151a1dd 50 . . . . . . . 8 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥))))
153152expcom 450 . . . . . . 7 (Lim 𝑥 → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))))
1545, 10, 15, 20, 23, 62, 153tfinds3 7230 . . . . . 6 (𝐵 ∈ On → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
155154com12 32 . . . . 5 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
156155adantrr 755 . . . 4 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
157156imp32 448 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
158157an32s 881 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
159 nnm0 7856 . . . 4 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
160159ad3antrrr 768 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 ∅) = ∅)
161 fnoe 7761 . . . . . . 7 𝑜 Fn (On × On)
162 fndm 6151 . . . . . . 7 ( ↑𝑜 Fn (On × On) → dom ↑𝑜 = (On × On))
163161, 162ax-mp 5 . . . . . 6 dom ↑𝑜 = (On × On)
164163ndmov 6984 . . . . 5 (¬ (ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑𝑜 𝐵) = ∅)
165164adantl 473 . . . 4 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (ω ↑𝑜 𝐵) = ∅)
166165oveq2d 6830 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (𝐴 ·𝑜 ∅))
167160, 166, 1653eqtr4d 2804 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
168158, 167pm2.61dan 867 1 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  wss 3715  c0 4058   ciun 4672   × cxp 5264  dom cdm 5266  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886   Fn wfn 6044  (class class class)co 6814  ωcom 7231  1𝑜c1o 7723  2𝑜c2o 7724   ·𝑜 comu 7728  𝑜 coe 7729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-omul 7735  df-oexp 7736
This theorem is referenced by:  cnfcom3  8776
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