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Theorem oeoalem 7920
Description: Lemma for oeoa 7921. (Contributed by Eric Schmidt, 26-May-2009.)
Hypotheses
Ref Expression
oeoalem.1 𝐴 ∈ On
oeoalem.2 ∅ ∈ 𝐴
oeoalem.3 𝐵 ∈ On
Assertion
Ref Expression
oeoalem (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))

Proof of Theorem oeoalem
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6889 . . . 4 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
21oveq2d 6897 . . 3 (𝑥 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 ∅)))
3 oveq2 6889 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
43oveq2d 6897 . . 3 (𝑥 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)))
52, 4eqeq12d 2832 . 2 (𝑥 = ∅ → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))))
6 oveq2 6889 . . . 4 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6897 . . 3 (𝑥 = 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝑦)))
8 oveq2 6889 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
98oveq2d 6897 . . 3 (𝑥 = 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
107, 9eqeq12d 2832 . 2 (𝑥 = 𝑦 → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))))
11 oveq2 6889 . . . 4 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1211oveq2d 6897 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)))
13 oveq2 6889 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
1413oveq2d 6897 . . 3 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
1512, 14eqeq12d 2832 . 2 (𝑥 = suc 𝑦 → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦))))
16 oveq2 6889 . . . 4 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1716oveq2d 6897 . . 3 (𝑥 = 𝐶 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝐶)))
18 oveq2 6889 . . . 4 (𝑥 = 𝐶 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐶))
1918oveq2d 6897 . . 3 (𝑥 = 𝐶 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
2017, 19eqeq12d 2832 . 2 (𝑥 = 𝐶 → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))))
21 oeoalem.1 . . . . 5 𝐴 ∈ On
22 oeoalem.3 . . . . 5 𝐵 ∈ On
23 oecl 7861 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
2421, 22, 23mp2an 675 . . . 4 (𝐴𝑜 𝐵) ∈ On
25 om1 7866 . . . 4 ((𝐴𝑜 𝐵) ∈ On → ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵)
27 oe0 7846 . . . . 5 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2821, 27ax-mp 5 . . . 4 (𝐴𝑜 ∅) = 1𝑜
2928oveq2i 6892 . . 3 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 1𝑜)
30 oa0 7840 . . . . 5 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
3122, 30ax-mp 5 . . . 4 (𝐵 +𝑜 ∅) = 𝐵
3231oveq2i 6892 . . 3 (𝐴𝑜 (𝐵 +𝑜 ∅)) = (𝐴𝑜 𝐵)
3326, 29, 323eqtr4ri 2850 . 2 (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))
34 oasuc 7848 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3534oveq2d 6897 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴𝑜 suc (𝐵 +𝑜 𝑦)))
36 oacl 7859 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
37 oesuc 7851 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3821, 36, 37sylancr 577 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3935, 38eqtrd 2851 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
4022, 39mpan 673 . . . . 5 (𝑦 ∈ On → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
41 oveq1 6888 . . . . 5 ((𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴) = (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴))
4240, 41sylan9eq 2871 . . . 4 ((𝑦 ∈ On ∧ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴))
43 oecl 7861 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
44 omass 7904 . . . . . . . . 9 (((𝐴𝑜 𝐵) ∈ On ∧ (𝐴𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
4524, 21, 44mp3an13 1569 . . . . . . . 8 ((𝐴𝑜 𝑦) ∈ On → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
4643, 45syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
47 oesuc 7851 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
4847oveq2d 6897 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
4946, 48eqtr4d 2854 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5021, 49mpan 673 . . . . 5 (𝑦 ∈ On → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5150adantr 468 . . . 4 ((𝑦 ∈ On ∧ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5242, 51eqtrd 2851 . . 3 ((𝑦 ∈ On ∧ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5352ex 399 . 2 (𝑦 ∈ On → ((𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦))))
54 vex 3405 . . . . . . . 8 𝑥 ∈ V
55 oalim 7856 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5622, 55mpan 673 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5754, 56mpan 673 . . . . . . 7 (Lim 𝑥 → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5857oveq2d 6897 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)))
5954a1i 11 . . . . . . 7 (Lim 𝑥𝑥 ∈ V)
60 limord 6007 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
61 ordelon 5971 . . . . . . . . . 10 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
6260, 61sylan 571 . . . . . . . . 9 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
6322, 62, 36sylancr 577 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
6463ralrimiva 3165 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On)
65 0ellim 6010 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
6665ne0d 4134 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
67 vex 3405 . . . . . . . . 9 𝑤 ∈ V
68 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
69 oelim 7858 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7068, 69mpan2 674 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7121, 70mpan 673 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7267, 71mpan 673 . . . . . . . 8 (Lim 𝑤 → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
73 oewordi 7915 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7468, 73mpan2 674 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7521, 74mp3an3 1567 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
76753impia 1138 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤))
7772, 76onoviun 7683 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
7859, 64, 66, 77syl3anc 1483 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
7958, 78eqtrd 2851 . . . . 5 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
80 iuneq2 4740 . . . . 5 (∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
8179, 80sylan9eq 2871 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
82 oelim 7858 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8368, 82mpan2 674 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8421, 83mpan 673 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8554, 84mpan 673 . . . . . . 7 (Lim 𝑥 → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8685oveq2d 6897 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)))
8721, 62, 43sylancr 577 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐴𝑜 𝑦) ∈ On)
8887ralrimiva 3165 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
89 omlim 7857 . . . . . . . . . 10 (((𝐴𝑜 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9024, 89mpan 673 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9167, 90mpan 673 . . . . . . . 8 (Lim 𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
92 omwordi 7895 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
9324, 92mp3an3 1567 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
94933impia 1138 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤))
9591, 94onoviun 7683 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9659, 88, 66, 95syl3anc 1483 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9786, 96eqtrd 2851 . . . . 5 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9897adantr 468 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9981, 98eqtr4d 2854 . . 3 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)))
10099ex 399 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥))))
1015, 10, 15, 20, 33, 53, 100tfinds 7296 1 (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2989  wral 3107  Vcvv 3402  wss 3780  c0 4127   ciun 4723  Ord word 5946  Oncon0 5947  Lim wlim 5948  suc csuc 5949  (class class class)co 6881  1𝑜c1o 7796   +𝑜 coa 7800   ·𝑜 comu 7801  𝑜 coe 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-om 7303  df-1st 7405  df-2nd 7406  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-1o 7803  df-2o 7804  df-oadd 7807  df-omul 7808  df-oexp 7809
This theorem is referenced by:  oeoa  7921
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