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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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