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Theorem oeoalem 7721
 Description: Lemma for oeoa 7722. (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 6698 . . . 4 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
21oveq2d 6706 . . 3 (𝑥 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 ∅)))
3 oveq2 6698 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
43oveq2d 6706 . . 3 (𝑥 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)))
52, 4eqeq12d 2666 . 2 (𝑥 = ∅ → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))))
6 oveq2 6698 . . . 4 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6706 . . 3 (𝑥 = 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝑦)))
8 oveq2 6698 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
98oveq2d 6706 . . 3 (𝑥 = 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
107, 9eqeq12d 2666 . 2 (𝑥 = 𝑦 → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))))
11 oveq2 6698 . . . 4 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1211oveq2d 6706 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)))
13 oveq2 6698 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
1413oveq2d 6706 . . 3 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
1512, 14eqeq12d 2666 . 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 6698 . . . 4 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1716oveq2d 6706 . . 3 (𝑥 = 𝐶 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝐶)))
18 oveq2 6698 . . . 4 (𝑥 = 𝐶 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐶))
1918oveq2d 6706 . . 3 (𝑥 = 𝐶 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
2017, 19eqeq12d 2666 . 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 7662 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
2421, 22, 23mp2an 708 . . . 4 (𝐴𝑜 𝐵) ∈ On
25 om1 7667 . . . 4 ((𝐴𝑜 𝐵) ∈ On → ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵)
27 oe0 7647 . . . . 5 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2821, 27ax-mp 5 . . . 4 (𝐴𝑜 ∅) = 1𝑜
2928oveq2i 6701 . . 3 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 1𝑜)
30 oa0 7641 . . . . 5 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
3122, 30ax-mp 5 . . . 4 (𝐵 +𝑜 ∅) = 𝐵
3231oveq2i 6701 . . 3 (𝐴𝑜 (𝐵 +𝑜 ∅)) = (𝐴𝑜 𝐵)
3326, 29, 323eqtr4ri 2684 . 2 (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))
34 oasuc 7649 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3534oveq2d 6706 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴𝑜 suc (𝐵 +𝑜 𝑦)))
36 oacl 7660 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
37 oesuc 7652 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3821, 36, 37sylancr 696 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3935, 38eqtrd 2685 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
4022, 39mpan 706 . . . . 5 (𝑦 ∈ On → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
41 oveq1 6697 . . . . 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 2705 . . . 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 7662 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
44 omass 7705 . . . . . . . . 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 1455 . . . . . . . 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 7652 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
4847oveq2d 6706 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
4946, 48eqtr4d 2688 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5021, 49mpan 706 . . . . 5 (𝑦 ∈ On → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5150adantr 480 . . . 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 2685 . . 3 ((𝑦 ∈ On ∧ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5352ex 449 . 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 3234 . . . . . . . 8 𝑥 ∈ V
55 oalim 7657 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5622, 55mpan 706 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5754, 56mpan 706 . . . . . . 7 (Lim 𝑥 → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5857oveq2d 6706 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)))
5954a1i 11 . . . . . . 7 (Lim 𝑥𝑥 ∈ V)
60 limord 5822 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
61 ordelon 5785 . . . . . . . . . 10 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
6260, 61sylan 487 . . . . . . . . 9 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
6322, 62, 36sylancr 696 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
6463ralrimiva 2995 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On)
65 0ellim 5825 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
66 ne0i 3954 . . . . . . . 8 (∅ ∈ 𝑥𝑥 ≠ ∅)
6765, 66syl 17 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
68 vex 3234 . . . . . . . . 9 𝑤 ∈ V
69 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
70 oelim 7659 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7169, 70mpan2 707 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7221, 71mpan 706 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7368, 72mpan 706 . . . . . . . 8 (Lim 𝑤 → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
74 oewordi 7716 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7569, 74mpan2 707 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7621, 75mp3an3 1453 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
77763impia 1280 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤))
7873, 77onoviun 7485 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
7959, 64, 67, 78syl3anc 1366 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
8058, 79eqtrd 2685 . . . . 5 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
81 iuneq2 4569 . . . . 5 (∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
8280, 81sylan9eq 2705 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
83 oelim 7659 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8469, 83mpan2 707 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8521, 84mpan 706 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8654, 85mpan 706 . . . . . . 7 (Lim 𝑥 → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8786oveq2d 6706 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)))
8821, 62, 43sylancr 696 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐴𝑜 𝑦) ∈ On)
8988ralrimiva 2995 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
90 omlim 7658 . . . . . . . . . 10 (((𝐴𝑜 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9124, 90mpan 706 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9268, 91mpan 706 . . . . . . . 8 (Lim 𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
93 omwordi 7696 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
9424, 93mp3an3 1453 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
95943impia 1280 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤))
9692, 95onoviun 7485 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9759, 89, 67, 96syl3anc 1366 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9887, 97eqtrd 2685 . . . . 5 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9998adantr 480 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
10082, 99eqtr4d 2688 . . 3 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)))
101100ex 449 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥))))
1025, 10, 15, 20, 33, 53, 101tfinds 7101 1 (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ∪ ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598   +𝑜 coa 7602   ·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:  oeoa  7722
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