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Theorem oeoalem 7621
Description: Lemma for oeoa 7622. (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 6612 . . . 4 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
21oveq2d 6620 . . 3 (𝑥 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 ∅)))
3 oveq2 6612 . . . 4 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
43oveq2d 6620 . . 3 (𝑥 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)))
52, 4eqeq12d 2636 . 2 (𝑥 = ∅ → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))))
6 oveq2 6612 . . . 4 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6620 . . 3 (𝑥 = 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝑦)))
8 oveq2 6612 . . . 4 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
98oveq2d 6620 . . 3 (𝑥 = 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
107, 9eqeq12d 2636 . 2 (𝑥 = 𝑦 → ((𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) ↔ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))))
11 oveq2 6612 . . . 4 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1211oveq2d 6620 . . 3 (𝑥 = suc 𝑦 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)))
13 oveq2 6612 . . . 4 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
1413oveq2d 6620 . . 3 (𝑥 = suc 𝑦 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
1512, 14eqeq12d 2636 . 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 6612 . . . 4 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1716oveq2d 6620 . . 3 (𝑥 = 𝐶 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 (𝐵 +𝑜 𝐶)))
18 oveq2 6612 . . . 4 (𝑥 = 𝐶 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐶))
1918oveq2d 6620 . . 3 (𝑥 = 𝐶 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
2017, 19eqeq12d 2636 . 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 7562 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
2421, 22, 23mp2an 707 . . . 4 (𝐴𝑜 𝐵) ∈ On
25 om1 7567 . . . 4 ((𝐴𝑜 𝐵) ∈ On → ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴𝑜 𝐵) ·𝑜 1𝑜) = (𝐴𝑜 𝐵)
27 oe0 7547 . . . . 5 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
2821, 27ax-mp 5 . . . 4 (𝐴𝑜 ∅) = 1𝑜
2928oveq2i 6615 . . 3 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 1𝑜)
30 oa0 7541 . . . . 5 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
3122, 30ax-mp 5 . . . 4 (𝐵 +𝑜 ∅) = 𝐵
3231oveq2i 6615 . . 3 (𝐴𝑜 (𝐵 +𝑜 ∅)) = (𝐴𝑜 𝐵)
3326, 29, 323eqtr4ri 2654 . 2 (𝐴𝑜 (𝐵 +𝑜 ∅)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 ∅))
34 oasuc 7549 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3534oveq2d 6620 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴𝑜 suc (𝐵 +𝑜 𝑦)))
36 oacl 7560 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
37 oesuc 7552 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3821, 36, 37sylancr 694 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
3935, 38eqtrd 2655 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
4022, 39mpan 705 . . . . 5 (𝑦 ∈ On → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 (𝐵 +𝑜 𝑦)) ·𝑜 𝐴))
41 oveq1 6611 . . . . 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 2675 . . . 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 7562 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
44 omass 7605 . . . . . . . . 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 1412 . . . . . . . 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 7552 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
4847oveq2d 6620 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
4946, 48eqtr4d 2658 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5021, 49mpan 705 . . . . 5 (𝑦 ∈ On → (((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5150adantr 481 . . . 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 2655 . . 3 ((𝑦 ∈ On ∧ (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 suc 𝑦)))
5352ex 450 . 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 3189 . . . . . . . 8 𝑥 ∈ V
55 oalim 7557 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5622, 55mpan 705 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5754, 56mpan 705 . . . . . . 7 (Lim 𝑥 → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
5857oveq2d 6620 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)))
5954a1i 11 . . . . . . 7 (Lim 𝑥𝑥 ∈ V)
60 limord 5743 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
61 ordelon 5706 . . . . . . . . . 10 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
6260, 61sylan 488 . . . . . . . . 9 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
6322, 62, 36sylancr 694 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
6463ralrimiva 2960 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On)
65 0ellim 5746 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
66 ne0i 3897 . . . . . . . 8 (∅ ∈ 𝑥𝑥 ≠ ∅)
6765, 66syl 17 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
68 vex 3189 . . . . . . . . 9 𝑤 ∈ V
69 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
70 oelim 7559 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7169, 70mpan2 706 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7221, 71mpan 705 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
7368, 72mpan 705 . . . . . . . 8 (Lim 𝑤 → (𝐴𝑜 𝑤) = 𝑧𝑤 (𝐴𝑜 𝑧))
74 oewordi 7616 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7569, 74mpan2 706 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
7621, 75mp3an3 1410 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤)))
77763impia 1258 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴𝑜 𝑧) ⊆ (𝐴𝑜 𝑤))
7873, 77onoviun 7385 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
7959, 64, 67, 78syl3anc 1323 . . . . . 6 (Lim 𝑥 → (𝐴𝑜 𝑦𝑥 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
8058, 79eqtrd 2655 . . . . 5 (Lim 𝑥 → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)))
81 iuneq2 4503 . . . . 5 (∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)) → 𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
8280, 81sylan9eq 2675 . . . 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 7559 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8469, 83mpan2 706 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8521, 84mpan 705 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8654, 85mpan 705 . . . . . . 7 (Lim 𝑥 → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
8786oveq2d 6620 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)))
8821, 62, 43sylancr 694 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐴𝑜 𝑦) ∈ On)
8988ralrimiva 2960 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On)
90 omlim 7558 . . . . . . . . . 10 (((𝐴𝑜 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9124, 90mpan 705 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
9268, 91mpan 705 . . . . . . . 8 (Lim 𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑤) = 𝑧𝑤 ((𝐴𝑜 𝐵) ·𝑜 𝑧))
93 omwordi 7596 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴𝑜 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
9424, 93mp3an3 1410 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤)))
95943impia 1258 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴𝑜 𝐵) ·𝑜 𝑧) ⊆ ((𝐴𝑜 𝐵) ·𝑜 𝑤))
9692, 95onoviun 7385 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴𝑜 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9759, 89, 67, 96syl3anc 1323 . . . . . 6 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 𝑦𝑥 (𝐴𝑜 𝑦)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9887, 97eqtrd 2655 . . . . 5 (Lim 𝑥 → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)) = 𝑦𝑥 ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦)))
9998adantr 481 . . . 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 2658 . . 3 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑦))) → (𝐴𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝑥)))
101100ex 450 . 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 7006 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 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  wss 3555  c0 3891   ciun 4485  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684  (class class class)co 6604  1𝑜c1o 7498   +𝑜 coa 7502   ·𝑜 comu 7503  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511
This theorem is referenced by:  oeoa  7622
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