MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeoalem Structured version   Visualization version   GIF version

Theorem oeoalem 8543
Description: Lemma for oeoa 8544. (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 7365 . . . 4 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
21oveq2d 7373 . . 3 (𝑥 = ∅ → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o ∅)))
3 oveq2 7365 . . . 4 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
43oveq2d 7373 . . 3 (𝑥 = ∅ → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o ∅)))
52, 4eqeq12d 2752 . 2 (𝑥 = ∅ → ((𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)) ↔ (𝐴o (𝐵 +o ∅)) = ((𝐴o 𝐵) ·o (𝐴o ∅))))
6 oveq2 7365 . . . 4 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7373 . . 3 (𝑥 = 𝑦 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o 𝑦)))
8 oveq2 7365 . . . 4 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
98oveq2d 7373 . . 3 (𝑥 = 𝑦 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
107, 9eqeq12d 2752 . 2 (𝑥 = 𝑦 → ((𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)) ↔ (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))))
11 oveq2 7365 . . . 4 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1211oveq2d 7373 . . 3 (𝑥 = suc 𝑦 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o suc 𝑦)))
13 oveq2 7365 . . . 4 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
1413oveq2d 7373 . . 3 (𝑥 = suc 𝑦 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
1512, 14eqeq12d 2752 . 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 7365 . . . 4 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1716oveq2d 7373 . . 3 (𝑥 = 𝐶 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o (𝐵 +o 𝐶)))
18 oveq2 7365 . . . 4 (𝑥 = 𝐶 → (𝐴o 𝑥) = (𝐴o 𝐶))
1918oveq2d 7373 . . 3 (𝑥 = 𝐶 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
2017, 19eqeq12d 2752 . 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 8483 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
2421, 22, 23mp2an 690 . . . 4 (𝐴o 𝐵) ∈ On
25 om1 8489 . . . 4 ((𝐴o 𝐵) ∈ On → ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵))
2624, 25ax-mp 5 . . 3 ((𝐴o 𝐵) ·o 1o) = (𝐴o 𝐵)
27 oe0 8468 . . . . 5 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2821, 27ax-mp 5 . . . 4 (𝐴o ∅) = 1o
2928oveq2i 7368 . . 3 ((𝐴o 𝐵) ·o (𝐴o ∅)) = ((𝐴o 𝐵) ·o 1o)
30 oa0 8462 . . . . 5 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
3122, 30ax-mp 5 . . . 4 (𝐵 +o ∅) = 𝐵
3231oveq2i 7368 . . 3 (𝐴o (𝐵 +o ∅)) = (𝐴o 𝐵)
3326, 29, 323eqtr4ri 2775 . 2 (𝐴o (𝐵 +o ∅)) = ((𝐴o 𝐵) ·o (𝐴o ∅))
34 oasuc 8470 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3534oveq2d 7373 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 +o suc 𝑦)) = (𝐴o suc (𝐵 +o 𝑦)))
36 oacl 8481 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
37 oesuc 8473 . . . . . . . 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 2776 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
4022, 39mpan 688 . . . . 5 (𝑦 ∈ On → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o (𝐵 +o 𝑦)) ·o 𝐴))
41 oveq1 7364 . . . . 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 2796 . . . 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 8483 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
44 omass 8527 . . . . . . . . 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 1452 . . . . . . . 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 8473 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
4847oveq2d 7373 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)) = ((𝐴o 𝐵) ·o ((𝐴o 𝑦) ·o 𝐴)))
4946, 48eqtr4d 2779 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (((𝐴o 𝐵) ·o (𝐴o 𝑦)) ·o 𝐴) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
5021, 49mpan 688 . . . . 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 2776 . . 3 ((𝑦 ∈ On ∧ (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))) → (𝐴o (𝐵 +o suc 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o suc 𝑦)))
5352ex 413 . 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 3449 . . . . . . . 8 𝑥 ∈ V
55 oalim 8478 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5622, 55mpan 688 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5754, 56mpan 688 . . . . . . 7 (Lim 𝑥 → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
5857oveq2d 7373 . . . . . 6 (Lim 𝑥 → (𝐴o (𝐵 +o 𝑥)) = (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)))
59 limord 6377 . . . . . . . . . 10 (Lim 𝑥 → Ord 𝑥)
60 ordelon 6341 . . . . . . . . . 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 3143 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On)
64 0ellim 6380 . . . . . . . 8 (Lim 𝑥 → ∅ ∈ 𝑥)
6564ne0d 4295 . . . . . . 7 (Lim 𝑥𝑥 ≠ ∅)
66 vex 3449 . . . . . . . . 9 𝑤 ∈ V
67 oeoalem.2 . . . . . . . . . . 11 ∅ ∈ 𝐴
68 oelim 8480 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6967, 68mpan2 689 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
7021, 69mpan 688 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
7166, 70mpan 688 . . . . . . . 8 (Lim 𝑤 → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
72 oewordi 8538 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7367, 72mpan2 689 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7421, 73mp3an3 1450 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
75743impia 1117 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴o 𝑧) ⊆ (𝐴o 𝑤))
7671, 75onoviun 8289 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 +o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7754, 63, 65, 76mp3an2i 1466 . . . . . 6 (Lim 𝑥 → (𝐴o 𝑦𝑥 (𝐵 +o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
7858, 77eqtrd 2776 . . . . 5 (Lim 𝑥 → (𝐴o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)))
79 iuneq2 4973 . . . . 5 (∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦)) → 𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
8078, 79sylan9eq 2796 . . . 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 8480 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8267, 81mpan2 689 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8321, 82mpan 688 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8454, 83mpan 688 . . . . . . 7 (Lim 𝑥 → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
8584oveq2d 7373 . . . . . 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 3143 . . . . . . 7 (Lim 𝑥 → ∀𝑦𝑥 (𝐴o 𝑦) ∈ On)
88 omlim 8479 . . . . . . . . . 10 (((𝐴o 𝐵) ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
8924, 88mpan 688 . . . . . . . . 9 ((𝑤 ∈ V ∧ Lim 𝑤) → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
9066, 89mpan 688 . . . . . . . 8 (Lim 𝑤 → ((𝐴o 𝐵) ·o 𝑤) = 𝑧𝑤 ((𝐴o 𝐵) ·o 𝑧))
91 omwordi 8518 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ (𝐴o 𝐵) ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
9224, 91mp3an3 1450 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤)))
93923impia 1117 . . . . . . . 8 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → ((𝐴o 𝐵) ·o 𝑧) ⊆ ((𝐴o 𝐵) ·o 𝑤))
9490, 93onoviun 8289 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9554, 87, 65, 94mp3an2i 1466 . . . . . 6 (Lim 𝑥 → ((𝐴o 𝐵) ·o 𝑦𝑥 (𝐴o 𝑦)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9685, 95eqtrd 2776 . . . . 5 (Lim 𝑥 → ((𝐴o 𝐵) ·o (𝐴o 𝑥)) = 𝑦𝑥 ((𝐴o 𝐵) ·o (𝐴o 𝑦)))
9796adantr 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)))
9880, 97eqtr4d 2779 . . 3 ((Lim 𝑥 ∧ ∀𝑦𝑥 (𝐴o (𝐵 +o 𝑦)) = ((𝐴o 𝐵) ·o (𝐴o 𝑦))) → (𝐴o (𝐵 +o 𝑥)) = ((𝐴o 𝐵) ·o (𝐴o 𝑥)))
9998ex 413 . 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 7796 1 (𝐶 ∈ On → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  wss 3910  c0 4282   ciun 4954  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1oc1o 8405   +o coa 8409   ·o comu 8410  o coe 8411
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418
This theorem is referenced by:  oeoa  8544
  Copyright terms: Public domain W3C validator