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Theorem oeoalem 7830
Description: Lemma for oeoa 7831. (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 6801 . . . 4 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
21oveq2d 6809 . . 3 (𝑥 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 ∅)))
3 oveq2 6801 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
43oveq2d 6809 . . 3 (𝑥 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)))
52, 4eqeq12d 2786 . 2 (𝑥 = ∅ → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))))
6 oveq2 6801 . . . 4 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6809 . . 3 (𝑥 = 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝑦)))
8 oveq2 6801 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
98oveq2d 6809 . . 3 (𝑥 = 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
107, 9eqeq12d 2786 . 2 (𝑥 = 𝑦 → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))))
11 oveq2 6801 . . . 4 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1211oveq2d 6809 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)))
13 oveq2 6801 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
1413oveq2d 6809 . . 3 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
1512, 14eqeq12d 2786 . 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 6801 . . . 4 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1716oveq2d 6809 . . 3 (𝑥 = 𝐶 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝐶)))
18 oveq2 6801 . . . 4 (𝑥 = 𝐶 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐶))
1918oveq2d 6809 . . 3 (𝑥 = 𝐶 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
2017, 19eqeq12d 2786 . 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 7771 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
2421, 22, 23mp2an 672 . . . 4 (𝐴𝑜 𝐵) ∈ On
25 om1 7776 . . . 4 ((𝐴𝑜 𝐵) ∈ On → ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵)
27 oe0 7756 . . . . 5 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2821, 27ax-mp 5 . . . 4 (𝐴𝑜 ∅) = 1𝑜
2928oveq2i 6804 . . 3 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 1𝑜)
30 oa0 7750 . . . . 5 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
3122, 30ax-mp 5 . . . 4 (𝐵 +𝑜 ∅) = 𝐵
3231oveq2i 6804 . . 3 (𝐴𝑜 (𝐵 +𝑜 ∅)) = (𝐴𝑜 𝐵)
3326, 29, 323eqtr4ri 2804 . 2 (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))
34 oasuc 7758 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3534oveq2d 6809 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴𝑜 suc (𝐵 +𝑜 𝑦)))
36 oacl 7769 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
37 oesuc 7761 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3821, 36, 37sylancr 575 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3935, 38eqtrd 2805 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
4022, 39mpan 670 . . . . 5 (𝑦 ∈ On → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
41 oveq1 6800 . . . . 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 2825 . . . 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 7771 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
44 omass 7814 . . . . . . . . 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 1563 . . . . . . . 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 7761 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
4847oveq2d 6809 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
4946, 48eqtr4d 2808 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5021, 49mpan 670 . . . . 5 (𝑦 ∈ On → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5150adantr 466 . . . 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 2805 . . 3 ((𝑦 ∈ On ∧ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5352ex 397 . 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 3354 . . . . . . . 8 𝑥 ∈ V
55 oalim 7766 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5622, 55mpan 670 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5754, 56mpan 670 . . . . . . 7 (Lim 𝑥 → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5857oveq2d 6809 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)))
5954a1i 11 . . . . . . 7 (Lim 𝑥𝑥 ∈ V)
60 limord 5927 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
61 ordelon 5890 . . . . . . . . . 10 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
6260, 61sylan 569 . . . . . . . . 9 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
6322, 62, 36sylancr 575 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
6463ralrimiva 3115 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On)
65 0ellim 5930 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
66 ne0i 4069 . . . . . . . 8 (∅ ∈ 𝑥𝑥 ≠ ∅)
6765, 66syl 17 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
68 vex 3354 . . . . . . . . 9 𝑤 ∈ V
69 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
70 oelim 7768 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7169, 70mpan2 671 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7221, 71mpan 670 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7368, 72mpan 670 . . . . . . . 8 (Lim 𝑤 → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
74 oewordi 7825 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7569, 74mpan2 671 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7621, 75mp3an3 1561 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
77763impia 1109 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤))
7873, 77onoviun 7593 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
7959, 64, 67, 78syl3anc 1476 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
8058, 79eqtrd 2805 . . . . 5 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
81 iuneq2 4671 . . . . 5 (∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
8280, 81sylan9eq 2825 . . . 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 7768 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8469, 83mpan2 671 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8521, 84mpan 670 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8654, 85mpan 670 . . . . . . 7 (Lim 𝑥 → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8786oveq2d 6809 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)))
8821, 62, 43sylancr 575 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐴𝑜 𝑦) ∈ On)
8988ralrimiva 3115 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
90 omlim 7767 . . . . . . . . . 10 (((𝐴𝑜 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9124, 90mpan 670 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9268, 91mpan 670 . . . . . . . 8 (Lim 𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
93 omwordi 7805 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
9424, 93mp3an3 1561 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
95943impia 1109 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤))
9692, 95onoviun 7593 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9759, 89, 67, 96syl3anc 1476 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9887, 97eqtrd 2805 . . . . 5 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9998adantr 466 . . . 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 2808 . . 3 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)))
101100ex 397 . 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 7206 1 (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  wss 3723  c0 4063   ciun 4654  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  (class class class)co 6793  1𝑜c1o 7706   +𝑜 coa 7710   ·𝑜 comu 7711  𝑜 coe 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719
This theorem is referenced by:  oeoa  7831
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