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Theorem oeoalem 8634
Description: Lemma for oeoa 8635. (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 7439 . . . 4 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
21oveq2d 7447 . . 3 (𝑥 = ∅ → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o ∅)))
3 oveq2 7439 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
43oveq2d 7447 . . 3 (𝑥 = ∅ → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o ∅)))
52, 4eqeq12d 2753 . 2 (𝑥 = ∅ → ((𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)) ↔ (𝐴o (𝐵 +o ∅)) = ((𝐴o 𝐵) ·o (𝐴o ∅))))
6 oveq2 7439 . . . 4 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7447 . . 3 (𝑥 = 𝑦 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o 𝑦)))
8 oveq2 7439 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
98oveq2d 7447 . . 3 (𝑥 = 𝑦 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
107, 9eqeq12d 2753 . 2 (𝑥 = 𝑦 → ((𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)) ↔ (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))))
11 oveq2 7439 . . . 4 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1211oveq2d 7447 . . 3 (𝑥 = suc 𝑦 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o suc 𝑦)))
13 oveq2 7439 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
1413oveq2d 7447 . . 3 (𝑥 = suc 𝑦 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
1512, 14eqeq12d 2753 . 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 7439 . . . 4 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1716oveq2d 7447 . . 3 (𝑥 = 𝐶 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o 𝐶)))
18 oveq2 7439 . . . 4 (𝑥 = 𝐶 → (𝐴o 𝑥) = (𝐴o 𝐶))
1918oveq2d 7447 . . 3 (𝑥 = 𝐶 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
2017, 19eqeq12d 2753 . 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 8575 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
2421, 22, 23mp2an 692 . . . 4 (𝐴o 𝐵) ∈ On
25 om1 8580 . . . 4 ((𝐴o 𝐵) ∈ On → ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵)
27 oe0 8560 . . . . 5 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2821, 27ax-mp 5 . . . 4 (𝐴o ∅) = 1o
2928oveq2i 7442 . . 3 ((𝐴o 𝐵) ·o (𝐴o ∅)) = ((𝐴o 𝐵) ·o 1o)
30 oa0 8554 . . . . 5 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
3122, 30ax-mp 5 . . . 4 (𝐵 +o ∅) = 𝐵
3231oveq2i 7442 . . 3 (𝐴o (𝐵 +o ∅)) = (𝐴o 𝐵)
3326, 29, 323eqtr4ri 2776 . 2 (𝐴o (𝐵 +o ∅)) = ((𝐴o 𝐵) ·o (𝐴o ∅))
34 oasuc 8562 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3534oveq2d 7447 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 +o suc 𝑦)) = (𝐴o suc (𝐵 +o 𝑦)))
36 oacl 8573 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
37 oesuc 8565 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴o suc (𝐵 +o 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
3821, 36, 37sylancr 587 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc (𝐵 +o 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
3935, 38eqtrd 2777 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
4022, 39mpan 690 . . . . 5 (𝑦 ∈ On → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
41 oveq1 7438 . . . . 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 2797 . . . 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 8575 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
44 omass 8618 . . . . . . . . 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 1454 . . . . . . . 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 8565 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
4847oveq2d 7447 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)) = ((𝐴o 𝐵) ·o ((𝐴o 𝑦) ·o 𝐴)))
4946, 48eqtr4d 2780 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴o 𝐵) ·o (𝐴o 𝑦)) ·o 𝐴) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
5021, 49mpan 690 . . . . 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 2777 . . 3 ((𝑦 ∈ On ∧ (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))) → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
5352ex 412 . 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 3484 . . . . . . . 8 𝑥 ∈ V
55 oalim 8570 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5622, 55mpan 690 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5754, 56mpan 690 . . . . . . 7 (Lim 𝑥 → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5857oveq2d 7447 . . . . . 6 (Lim 𝑥 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)))
59 limord 6444 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
60 ordelon 6408 . . . . . . . . . 10 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
6159, 60sylan 580 . . . . . . . . 9 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
6222, 61, 36sylancr 587 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐵 +o 𝑦) ∈ On)
6362ralrimiva 3146 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On)
64 0ellim 6447 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
6564ne0d 4342 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
66 vex 3484 . . . . . . . . 9 𝑤 ∈ V
67 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
68 oelim 8572 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6967, 68mpan2 691 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
7021, 69mpan 690 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
7166, 70mpan 690 . . . . . . . 8 (Lim 𝑤 → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
72 oewordi 8629 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7367, 72mpan2 691 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7421, 73mp3an3 1452 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
75743impia 1118 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴o 𝑧) ⊆ (𝐴o 𝑤))
7671, 75onoviun 8383 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7754, 63, 65, 76mp3an2i 1468 . . . . . 6 (Lim 𝑥 → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7858, 77eqtrd 2777 . . . . 5 (Lim 𝑥 → (𝐴o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
79 iuneq2 5011 . . . . 5 (∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)) → 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
8078, 79sylan9eq 2797 . . . 4 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))) → (𝐴o (𝐵 +o 𝑥)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
81 oelim 8572 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8267, 81mpan2 691 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8321, 82mpan 690 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8454, 83mpan 690 . . . . . . 7 (Lim 𝑥 → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8584oveq2d 7447 . . . . . 6 (Lim 𝑥 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)))
8621, 61, 43sylancr 587 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐴o 𝑦) ∈ On)
8786ralrimiva 3146 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴o 𝑦) ∈ On)
88 omlim 8571 . . . . . . . . . 10 (((𝐴o 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
8924, 88mpan 690 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
9066, 89mpan 690 . . . . . . . 8 (Lim 𝑤 → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
91 omwordi 8609 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴o 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
9224, 91mp3an3 1452 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
93923impia 1118 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤))
9490, 93onoviun 8383 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9554, 87, 65, 94mp3an2i 1468 . . . . . 6 (Lim 𝑥 → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9685, 95eqtrd 2777 . . . . 5 (Lim 𝑥 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9796adantr 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)))
9880, 97eqtr4d 2780 . . 3 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))) → (𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)))
9998ex 412 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)) → (𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥))))
1005, 10, 15, 20, 33, 53, 99tfinds 7881 1 (𝐶 ∈ On → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  wss 3951  c0 4333   ciun 4991  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431  1oc1o 8499   +o coa 8503   ·o comu 8504  o coe 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by:  oeoa  8635
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