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Theorem omabs 7772
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 2719 . . . . . . . 8 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2 oveq2 6698 . . . . . . . . . 10 (𝑥 = ∅ → (ω ↑𝑜 𝑥) = (ω ↑𝑜 ∅))
32oveq2d 6706 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 ∅)))
43, 2eqeq12d 2666 . . . . . . . 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 2719 . . . . . . . 8 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
7 oveq2 6698 . . . . . . . . . 10 (𝑥 = 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦))
87oveq2d 6706 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝑦)))
98, 7eqeq12d 2666 . . . . . . . 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 2719 . . . . . . . 8 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
12 oveq2 6698 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 suc 𝑦))
1312oveq2d 6706 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)))
1413, 12eqeq12d 2666 . . . . . . . 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 2719 . . . . . . . 8 (𝑥 = 𝐵 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐵))
17 oveq2 6698 . . . . . . . . . 10 (𝑥 = 𝐵 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝐵))
1817oveq2d 6706 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝐵)))
1918, 17eqeq12d 2666 . . . . . . . 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 3952 . . . . . . . . 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 809 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ω ∈ On)
25 simpll 805 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ ω)
26 simplr 807 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ∅ ∈ 𝐴)
27 omabslem 7771 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω)
2824, 25, 26, 27syl3anc 1366 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 ω) = ω)
2928adantr 480 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 ω) = ω)
30 suceq 5828 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → suc 𝑦 = suc ∅)
31 df-1o 7605 . . . . . . . . . . . . . . . . . 18 1𝑜 = suc ∅
3230, 31syl6eqr 2703 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → suc 𝑦 = 1𝑜)
3332oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (ω ↑𝑜 suc 𝑦) = (ω ↑𝑜 1𝑜))
34 oe1 7669 . . . . . . . . . . . . . . . . 17 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
3534ad2antrl 764 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 1𝑜) = ω)
3633, 35sylan9eqr 2707 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (ω ↑𝑜 suc 𝑦) = ω)
3736oveq2d 6706 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ω))
3829, 37, 363eqtr4d 2695 . . . . . . . . . . . . 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 6697 . . . . . . . . . . . . . 14 ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
42 oesuc 7652 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
4342adantl 481 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
4443oveq2d 6706 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
45 nnon 7113 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ω → 𝐴 ∈ On)
4645ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ On)
47 oecl 7662 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 𝑦) ∈ On)
4847adantl 481 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 𝑦) ∈ On)
49 omass 7705 . . . . . . . . . . . . . . . . 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 1366 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
5144, 50eqtr4d 2688 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω))
5251, 43eqeq12d 2666 . . . . . . . . . . . . . 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 811 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝑦 ∈ On)
57 on0eqel 5883 . . . . . . . . . . . 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 681 . . . . . . . 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 766 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝐴 ∈ On)
64 simprl 809 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ On)
65 simprr 811 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → Lim 𝑥)
66 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
6765, 66jctil 559 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (𝑥 ∈ V ∧ Lim 𝑥))
68 limelon 5826 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
6967, 68syl 17 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝑥 ∈ On)
70 oecl 7662 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ↑𝑜 𝑥) ∈ On)
7164, 69, 70syl2anc 694 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) ∈ On)
7271adantr 480 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ∈ On)
73 1onn 7764 . . . . . . . . . . . . . . . . . 18 1𝑜 ∈ ω
7473a1i 11 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 1𝑜 ∈ ω)
75 ondif2 7627 . . . . . . . . . . . . . . . . 17 (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
7664, 74, 75sylanbrc 699 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ (On ∖ 2𝑜))
7776adantr 480 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ (On ∖ 2𝑜))
7867adantr 480 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑥 ∈ V ∧ Lim 𝑥))
79 oelimcl 7725 . . . . . . . . . . . . . . 15 ((ω ∈ (On ∖ 2𝑜) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (ω ↑𝑜 𝑥))
8077, 78, 79syl2anc 694 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → Lim (ω ↑𝑜 𝑥))
81 omlim 7658 . . . . . . . . . . . . . 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 1364 . . . . . . . . . . . . 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 817 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ On)
84 oelim2 7720 . . . . . . . . . . . . . . . . . . . 20 ((ω ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) = 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦))
8583, 78, 84syl2anc 694 . . . . . . . . . . . . . . . . . . 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 2716 . . . . . . . . . . . . . . . . . 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 4556 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝑥 ∈ On)
90 anass 682 . . . . . . . . . . . . . . . . . . . 20 (((𝑦𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ (𝑦𝑥 ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))))
91 onelon 5786 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
92 on0eln0 5818 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (∅ ∈ 𝑦𝑦 ≠ ∅))
9391, 92syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ On ∧ 𝑦𝑥) → (∅ ∈ 𝑦𝑦 ≠ ∅))
9493pm5.32da 674 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ (𝑦𝑥𝑦 ≠ ∅)))
95 dif1o 7625 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝑥 ∖ 1𝑜) ↔ (𝑦𝑥𝑦 ≠ ∅))
9694, 95syl6bbr 278 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ 𝑦 ∈ (𝑥 ∖ 1𝑜)))
9796anbi1d 741 . . . . . . . . . . . . . . . . . . . 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 3077 . . . . . . . . . . . . . . . . . 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 3101 . . . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . . . . . . . . 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 680 . . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . . . 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 5771 . . . . . . . . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ω ∈ On)
11269ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5786 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((ω ↑𝑜 𝑦) ∈ On ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → 𝑧 ∈ On)
117115, 110, 116syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑧 ∈ On)
11845ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝐴 ∈ On)
119118ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ∅ ∈ 𝐴)
121120ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
122 omord2 7692 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . . . . . 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 7713 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ ω ∈ (On ∖ 2𝑜)) → (𝑦𝑥 ↔ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)))
129114, 112, 127, 128syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . 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 5809 . . . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . . 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 5777 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord (ω ↑𝑜 𝑥) ∧ (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
135109, 133, 134syl2anc 694 . . . . . . . . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . . . . . . 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 2994 . . . . . . . . . . . . . 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 4593 . . . . . . . . . . . . . 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 3672 . . . . . . . . . . . 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 815 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
147 omword2 7699 . . . . . . . . . . . . 13 ((((ω ↑𝑜 𝑥) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (ω ↑𝑜 𝑥) ⊆ (𝐴 ·𝑜 (ω ↑𝑜 𝑥)))
14872, 63, 146, 147syl21anc 1365 . . . . . . . . . . . 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 3653 . . . . . . . . . . 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 681 . . . . . . . . 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 7106 . . . . . 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 753 . . . 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 863 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
159 nnm0 7730 . . . 4 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
160159ad3antrrr 766 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 ∅) = ∅)
161 fnoe 7635 . . . . . . 7 𝑜 Fn (On × On)
162 fndm 6028 . . . . . . 7 ( ↑𝑜 Fn (On × On) → dom ↑𝑜 = (On × On))
163161, 162ax-mp 5 . . . . . 6 dom ↑𝑜 = (On × On)
164163ndmov 6860 . . . . 5 (¬ (ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑𝑜 𝐵) = ∅)
165164adantl 481 . . . 4 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (ω ↑𝑜 𝐵) = ∅)
166165oveq2d 6706 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (𝐴 ·𝑜 ∅))
167160, 166, 1653eqtr4d 2695 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
168158, 167pm2.61dan 849 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 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607  c0 3948   ciun 4552   × cxp 5141  dom cdm 5143  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  (class class class)co 6690  ωcom 7107  1𝑜c1o 7598  2𝑜c2o 7599   ·𝑜 comu 7603  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  cnfcom3  8639
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