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Theorem oeoalem 8633
Description: Lemma for oeoa 8634. (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 2751 . 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 2751 . 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 2751 . 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 2751 . 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 8574 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
2421, 22, 23mp2an 692 . . . 4 (𝐴o 𝐵) ∈ On
25 om1 8579 . . . 4 ((𝐴o 𝐵) ∈ On → ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵)
27 oe0 8559 . . . . 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 8553 . . . . 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 2774 . 2 (𝐴o (𝐵 +o ∅)) = ((𝐴o 𝐵) ·o (𝐴o ∅))
34 oasuc 8561 . . . . . . . 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 8572 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
37 oesuc 8564 . . . . . . . 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 2775 . . . . . 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 2795 . . . 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 8574 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
44 omass 8617 . . . . . . . . 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 1451 . . . . . . . 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 8564 . . . . . . . 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 2778 . . . . . 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 2775 . . 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 3482 . . . . . . . 8 𝑥 ∈ V
55 oalim 8569 . . . . . . . . 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 6446 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
60 ordelon 6410 . . . . . . . . . 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 3144 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On)
64 0ellim 6449 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
6564ne0d 4348 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
66 vex 3482 . . . . . . . . 9 𝑤 ∈ V
67 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
68 oelim 8571 . . . . . . . . . . 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 8628 . . . . . . . . . . 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 1449 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
75743impia 1116 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴o 𝑧) ⊆ (𝐴o 𝑤))
7671, 75onoviun 8382 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7754, 63, 65, 76mp3an2i 1465 . . . . . 6 (Lim 𝑥 → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7858, 77eqtrd 2775 . . . . 5 (Lim 𝑥 → (𝐴o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
79 iuneq2 5016 . . . . 5 (∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)) → 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
8078, 79sylan9eq 2795 . . . 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 8571 . . . . . . . . . 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 3144 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴o 𝑦) ∈ On)
88 omlim 8570 . . . . . . . . . 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 8608 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴o 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
9224, 91mp3an3 1449 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
93923impia 1116 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤))
9490, 93onoviun 8382 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9554, 87, 65, 94mp3an2i 1465 . . . . . 6 (Lim 𝑥 → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9685, 95eqtrd 2775 . . . . 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 2778 . . 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 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478  wss 3963  c0 4339   ciun 4996  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431  1oc1o 8498   +o coa 8502   ·o comu 8503  o coe 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511
This theorem is referenced by:  oeoa  8634
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