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Theorem oeoelem 8545
Description: Lemma for oeoe 8546. (Contributed by Eric Schmidt, 26-May-2009.)
Hypotheses
Ref Expression
oeoelem.1 𝐴 ∈ On
oeoelem.2 ∅ ∈ 𝐴
Assertion
Ref Expression
oeoelem ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))

Proof of Theorem oeoelem
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7365 . . . 4 (𝑥 = ∅ → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o ∅))
2 oveq2 7365 . . . . 5 (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
32oveq2d 7373 . . . 4 (𝑥 = ∅ → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o ∅)))
41, 3eqeq12d 2752 . . 3 (𝑥 = ∅ → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ ((𝐴o 𝐵) ↑o ∅) = (𝐴o (𝐵 ·o ∅))))
5 oveq2 7365 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o 𝑦))
6 oveq2 7365 . . . . 5 (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
76oveq2d 7373 . . . 4 (𝑥 = 𝑦 → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o 𝑦)))
85, 7eqeq12d 2752 . . 3 (𝑥 = 𝑦 → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ ((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦))))
9 oveq2 7365 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o suc 𝑦))
10 oveq2 7365 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
1110oveq2d 7373 . . . 4 (𝑥 = suc 𝑦 → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o suc 𝑦)))
129, 11eqeq12d 2752 . . 3 (𝑥 = suc 𝑦 → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ ((𝐴o 𝐵) ↑o suc 𝑦) = (𝐴o (𝐵 ·o suc 𝑦))))
13 oveq2 7365 . . . 4 (𝑥 = 𝐶 → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o 𝐶))
14 oveq2 7365 . . . . 5 (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
1514oveq2d 7373 . . . 4 (𝑥 = 𝐶 → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o 𝐶)))
1613, 15eqeq12d 2752 . . 3 (𝑥 = 𝐶 → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶))))
17 oeoelem.1 . . . . . 6 𝐴 ∈ On
18 oecl 8483 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
1917, 18mpan 688 . . . . 5 (𝐵 ∈ On → (𝐴o 𝐵) ∈ On)
20 oe0 8468 . . . . 5 ((𝐴o 𝐵) ∈ On → ((𝐴o 𝐵) ↑o ∅) = 1o)
2119, 20syl 17 . . . 4 (𝐵 ∈ On → ((𝐴o 𝐵) ↑o ∅) = 1o)
22 om0 8463 . . . . . 6 (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
2322oveq2d 7373 . . . . 5 (𝐵 ∈ On → (𝐴o (𝐵 ·o ∅)) = (𝐴o ∅))
24 oe0 8468 . . . . . 6 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2517, 24ax-mp 5 . . . . 5 (𝐴o ∅) = 1o
2623, 25eqtrdi 2792 . . . 4 (𝐵 ∈ On → (𝐴o (𝐵 ·o ∅)) = 1o)
2721, 26eqtr4d 2779 . . 3 (𝐵 ∈ On → ((𝐴o 𝐵) ↑o ∅) = (𝐴o (𝐵 ·o ∅)))
28 oveq1 7364 . . . . 5 (((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦)) → (((𝐴o 𝐵) ↑o 𝑦) ·o (𝐴o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
29 oesuc 8473 . . . . . . 7 (((𝐴o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝐵) ↑o suc 𝑦) = (((𝐴o 𝐵) ↑o 𝑦) ·o (𝐴o 𝐵)))
3019, 29sylan 580 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴o 𝐵) ↑o suc 𝑦) = (((𝐴o 𝐵) ↑o 𝑦) ·o (𝐴o 𝐵)))
31 omsuc 8472 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
3231oveq2d 7373 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 ·o suc 𝑦)) = (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)))
33 omcl 8482 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
34 oeoa 8544 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3517, 34mp3an1 1448 . . . . . . . . 9 (((𝐵 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3633, 35sylan 580 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐵 ∈ On) → (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3736anabss1 664 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3832, 37eqtrd 2776 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 ·o suc 𝑦)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3930, 38eqeq12d 2752 . . . . 5 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴o 𝐵) ↑o suc 𝑦) = (𝐴o (𝐵 ·o suc 𝑦)) ↔ (((𝐴o 𝐵) ↑o 𝑦) ·o (𝐴o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵))))
4028, 39syl5ibr 245 . . . 4 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦)) → ((𝐴o 𝐵) ↑o suc 𝑦) = (𝐴o (𝐵 ·o suc 𝑦))))
4140expcom 414 . . 3 (𝑦 ∈ On → (𝐵 ∈ On → (((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦)) → ((𝐴o 𝐵) ↑o suc 𝑦) = (𝐴o (𝐵 ·o suc 𝑦)))))
42 iuneq2 4973 . . . . 5 (∀𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦)) → 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
43 vex 3449 . . . . . . 7 𝑥 ∈ V
44 oeoelem.2 . . . . . . . . . 10 ∅ ∈ 𝐴
45 oen0 8533 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))
4644, 45mpan2 689 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴o 𝐵))
47 oelim 8480 . . . . . . . . . 10 ((((𝐴o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴o 𝐵)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
4818, 47sylanl1 678 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴o 𝐵)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
4946, 48mpidan 687 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
5017, 49mpanl1 698 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
5143, 50mpanr1 701 . . . . . 6 ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
52 omlim 8479 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = 𝑦𝑥 (𝐵 ·o 𝑦))
5343, 52mpanr1 701 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) = 𝑦𝑥 (𝐵 ·o 𝑦))
5453oveq2d 7373 . . . . . . 7 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o 𝑦𝑥 (𝐵 ·o 𝑦)))
55 limord 6377 . . . . . . . . . . . 12 (Lim 𝑥 → Ord 𝑥)
56 ordelon 6341 . . . . . . . . . . . 12 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
5755, 56sylan 580 . . . . . . . . . . 11 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
5857, 33sylan2 593 . . . . . . . . . 10 ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐵 ·o 𝑦) ∈ On)
5958anassrs 468 . . . . . . . . 9 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦𝑥) → (𝐵 ·o 𝑦) ∈ On)
6059ralrimiva 3143 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦𝑥 (𝐵 ·o 𝑦) ∈ On)
61 0ellim 6380 . . . . . . . . . 10 (Lim 𝑥 → ∅ ∈ 𝑥)
6261ne0d 4295 . . . . . . . . 9 (Lim 𝑥𝑥 ≠ ∅)
6362adantl 482 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
64 vex 3449 . . . . . . . . . 10 𝑤 ∈ V
65 oelim 8480 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6644, 65mpan2 689 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6717, 66mpan 688 . . . . . . . . . 10 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6864, 67mpan 688 . . . . . . . . 9 (Lim 𝑤 → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
69 oewordi 8538 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7044, 69mpan2 689 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7117, 70mp3an3 1450 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
72713impia 1117 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴o 𝑧) ⊆ (𝐴o 𝑤))
7368, 72onoviun 8289 . . . . . . . 8 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 ·o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴o 𝑦𝑥 (𝐵 ·o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
7443, 60, 63, 73mp3an2i 1466 . . . . . . 7 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴o 𝑦𝑥 (𝐵 ·o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
7554, 74eqtrd 2776 . . . . . 6 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴o (𝐵 ·o 𝑥)) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
7651, 75eqeq12d 2752 . . . . 5 ((𝐵 ∈ On ∧ Lim 𝑥) → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦))))
7742, 76syl5ibr 245 . . . 4 ((𝐵 ∈ On ∧ Lim 𝑥) → (∀𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦)) → ((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥))))
7877expcom 414 . . 3 (Lim 𝑥 → (𝐵 ∈ On → (∀𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦)) → ((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)))))
794, 8, 12, 16, 27, 41, 78tfinds3 7801 . 2 (𝐶 ∈ On → (𝐵 ∈ On → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶))))
8079impcom 408 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 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:  oeoe  8546
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