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Theorem oeoalem 8205
 Description: Lemma for oeoa 8206. (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 7143 . . . 4 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
21oveq2d 7151 . . 3 (𝑥 = ∅ → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o ∅)))
3 oveq2 7143 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
43oveq2d 7151 . . 3 (𝑥 = ∅ → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o ∅)))
52, 4eqeq12d 2814 . 2 (𝑥 = ∅ → ((𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)) ↔ (𝐴o (𝐵 +o ∅)) = ((𝐴o 𝐵) ·o (𝐴o ∅))))
6 oveq2 7143 . . . 4 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7151 . . 3 (𝑥 = 𝑦 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o 𝑦)))
8 oveq2 7143 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
98oveq2d 7151 . . 3 (𝑥 = 𝑦 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
107, 9eqeq12d 2814 . 2 (𝑥 = 𝑦 → ((𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)) ↔ (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))))
11 oveq2 7143 . . . 4 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1211oveq2d 7151 . . 3 (𝑥 = suc 𝑦 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o suc 𝑦)))
13 oveq2 7143 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
1413oveq2d 7151 . . 3 (𝑥 = suc 𝑦 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
1512, 14eqeq12d 2814 . 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 7143 . . . 4 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1716oveq2d 7151 . . 3 (𝑥 = 𝐶 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o 𝐶)))
18 oveq2 7143 . . . 4 (𝑥 = 𝐶 → (𝐴o 𝑥) = (𝐴o 𝐶))
1918oveq2d 7151 . . 3 (𝑥 = 𝐶 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
2017, 19eqeq12d 2814 . 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 8145 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
2421, 22, 23mp2an 691 . . . 4 (𝐴o 𝐵) ∈ On
25 om1 8151 . . . 4 ((𝐴o 𝐵) ∈ On → ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵)
27 oe0 8130 . . . . 5 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2821, 27ax-mp 5 . . . 4 (𝐴o ∅) = 1o
2928oveq2i 7146 . . 3 ((𝐴o 𝐵) ·o (𝐴o ∅)) = ((𝐴o 𝐵) ·o 1o)
30 oa0 8124 . . . . 5 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
3122, 30ax-mp 5 . . . 4 (𝐵 +o ∅) = 𝐵
3231oveq2i 7146 . . 3 (𝐴o (𝐵 +o ∅)) = (𝐴o 𝐵)
3326, 29, 323eqtr4ri 2832 . 2 (𝐴o (𝐵 +o ∅)) = ((𝐴o 𝐵) ·o (𝐴o ∅))
34 oasuc 8132 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3534oveq2d 7151 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 +o suc 𝑦)) = (𝐴o suc (𝐵 +o 𝑦)))
36 oacl 8143 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
37 oesuc 8135 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴o suc (𝐵 +o 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
3821, 36, 37sylancr 590 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc (𝐵 +o 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
3935, 38eqtrd 2833 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
4022, 39mpan 689 . . . . 5 (𝑦 ∈ On → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
41 oveq1 7142 . . . . 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 2853 . . . 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 8145 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
44 omass 8189 . . . . . . . . 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 1449 . . . . . . . 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 8135 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
4847oveq2d 7151 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)) = ((𝐴o 𝐵) ·o ((𝐴o 𝑦) ·o 𝐴)))
4946, 48eqtr4d 2836 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴o 𝐵) ·o (𝐴o 𝑦)) ·o 𝐴) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
5021, 49mpan 689 . . . . 5 (𝑦 ∈ On → (((𝐴o 𝐵) ·o (𝐴o 𝑦)) ·o 𝐴) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
5150adantr 484 . . . 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 2833 . . 3 ((𝑦 ∈ On ∧ (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))) → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
5352ex 416 . 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 3444 . . . . . . . 8 𝑥 ∈ V
55 oalim 8140 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5622, 55mpan 689 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5754, 56mpan 689 . . . . . . 7 (Lim 𝑥 → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5857oveq2d 7151 . . . . . 6 (Lim 𝑥 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)))
59 limord 6218 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
60 ordelon 6183 . . . . . . . . . 10 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
6159, 60sylan 583 . . . . . . . . 9 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
6222, 61, 36sylancr 590 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐵 +o 𝑦) ∈ On)
6362ralrimiva 3149 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On)
64 0ellim 6221 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
6564ne0d 4251 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
66 vex 3444 . . . . . . . . 9 𝑤 ∈ V
67 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
68 oelim 8142 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6967, 68mpan2 690 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
7021, 69mpan 689 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
7166, 70mpan 689 . . . . . . . 8 (Lim 𝑤 → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
72 oewordi 8200 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7367, 72mpan2 690 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7421, 73mp3an3 1447 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
75743impia 1114 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴o 𝑧) ⊆ (𝐴o 𝑤))
7671, 75onoviun 7963 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7754, 63, 65, 76mp3an2i 1463 . . . . . 6 (Lim 𝑥 → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7858, 77eqtrd 2833 . . . . 5 (Lim 𝑥 → (𝐴o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
79 iuneq2 4900 . . . . 5 (∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)) → 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
8078, 79sylan9eq 2853 . . . 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 8142 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8267, 81mpan2 690 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8321, 82mpan 689 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8454, 83mpan 689 . . . . . . 7 (Lim 𝑥 → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8584oveq2d 7151 . . . . . 6 (Lim 𝑥 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)))
8621, 61, 43sylancr 590 . . . . . . . 8 ((Lim 𝑥𝑦𝑥) → (𝐴o 𝑦) ∈ On)
8786ralrimiva 3149 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴o 𝑦) ∈ On)
88 omlim 8141 . . . . . . . . . 10 (((𝐴o 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
8924, 88mpan 689 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
9066, 89mpan 689 . . . . . . . 8 (Lim 𝑤 → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
91 omwordi 8180 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴o 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
9224, 91mp3an3 1447 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
93923impia 1114 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤))
9490, 93onoviun 7963 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9554, 87, 65, 94mp3an2i 1463 . . . . . 6 (Lim 𝑥 → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9685, 95eqtrd 2833 . . . . 5 (Lim 𝑥 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9796adantr 484 . . . 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 2836 . . 3 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))) → (𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)))
9998ex 416 . 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 7554 1 (𝐶 ∈ On → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∪ ciun 4881  Ord word 6158  Oncon0 6159  Lim wlim 6160  suc csuc 6161  (class class class)co 7135  1oc1o 8078   +o coa 8082   ·o comu 8083   ↑o coe 8084 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-oexp 8091 This theorem is referenced by:  oeoa  8206
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