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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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