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Theorem oeoelem 8213
Description: Lemma for oeoe 8214. (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 7153 . . . 4 (𝑥 = ∅ → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o ∅))
2 oveq2 7153 . . . . 5 (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
32oveq2d 7161 . . . 4 (𝑥 = ∅ → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o ∅)))
41, 3eqeq12d 2834 . . 3 (𝑥 = ∅ → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ ((𝐴o 𝐵) ↑o ∅) = (𝐴o (𝐵 ·o ∅))))
5 oveq2 7153 . . . 4 (𝑥 = 𝑦 → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o 𝑦))
6 oveq2 7153 . . . . 5 (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
76oveq2d 7161 . . . 4 (𝑥 = 𝑦 → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o 𝑦)))
85, 7eqeq12d 2834 . . 3 (𝑥 = 𝑦 → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ ((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦))))
9 oveq2 7153 . . . 4 (𝑥 = suc 𝑦 → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o suc 𝑦))
10 oveq2 7153 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
1110oveq2d 7161 . . . 4 (𝑥 = suc 𝑦 → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o suc 𝑦)))
129, 11eqeq12d 2834 . . 3 (𝑥 = suc 𝑦 → (((𝐴o 𝐵) ↑o 𝑥) = (𝐴o (𝐵 ·o 𝑥)) ↔ ((𝐴o 𝐵) ↑o suc 𝑦) = (𝐴o (𝐵 ·o suc 𝑦))))
13 oveq2 7153 . . . 4 (𝑥 = 𝐶 → ((𝐴o 𝐵) ↑o 𝑥) = ((𝐴o 𝐵) ↑o 𝐶))
14 oveq2 7153 . . . . 5 (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
1514oveq2d 7161 . . . 4 (𝑥 = 𝐶 → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o (𝐵 ·o 𝐶)))
1613, 15eqeq12d 2834 . . 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 8151 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
1917, 18mpan 686 . . . . 5 (𝐵 ∈ On → (𝐴o 𝐵) ∈ On)
20 oe0 8136 . . . . 5 ((𝐴o 𝐵) ∈ On → ((𝐴o 𝐵) ↑o ∅) = 1o)
2119, 20syl 17 . . . 4 (𝐵 ∈ On → ((𝐴o 𝐵) ↑o ∅) = 1o)
22 om0 8131 . . . . . 6 (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
2322oveq2d 7161 . . . . 5 (𝐵 ∈ On → (𝐴o (𝐵 ·o ∅)) = (𝐴o ∅))
24 oe0 8136 . . . . . 6 (𝐴 ∈ On → (𝐴o ∅) = 1o)
2517, 24ax-mp 5 . . . . 5 (𝐴o ∅) = 1o
2623, 25syl6eq 2869 . . . 4 (𝐵 ∈ On → (𝐴o (𝐵 ·o ∅)) = 1o)
2721, 26eqtr4d 2856 . . 3 (𝐵 ∈ On → ((𝐴o 𝐵) ↑o ∅) = (𝐴o (𝐵 ·o ∅)))
28 oveq1 7152 . . . . 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 8141 . . . . . . 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 8140 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
3231oveq2d 7161 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 ·o suc 𝑦)) = (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)))
33 omcl 8150 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
34 oeoa 8212 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3517, 34mp3an1 1439 . . . . . . . . 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 662 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3832, 37eqtrd 2853 . . . . . 6 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴o (𝐵 ·o suc 𝑦)) = ((𝐴o (𝐵 ·o 𝑦)) ·o (𝐴o 𝐵)))
3930, 38eqeq12d 2834 . . . . 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 247 . . . 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 4929 . . . . 5 (∀𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦) = (𝐴o (𝐵 ·o 𝑦)) → 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
43 vex 3495 . . . . . . 7 𝑥 ∈ V
44 oeoelem.2 . . . . . . . . . 10 ∅ ∈ 𝐴
45 oen0 8201 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))
4644, 45mpan2 687 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∅ ∈ (𝐴o 𝐵))
47 oelim 8148 . . . . . . . . . 10 ((((𝐴o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴o 𝐵)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
4818, 47sylanl1 676 . . . . . . . . 9 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ (𝐴o 𝐵)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
4946, 48mpidan 685 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
5017, 49mpanl1 696 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
5143, 50mpanr1 699 . . . . . 6 ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐴o 𝐵) ↑o 𝑥) = 𝑦𝑥 ((𝐴o 𝐵) ↑o 𝑦))
52 omlim 8147 . . . . . . . . 9 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·o 𝑥) = 𝑦𝑥 (𝐵 ·o 𝑦))
5343, 52mpanr1 699 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) = 𝑦𝑥 (𝐵 ·o 𝑦))
5453oveq2d 7161 . . . . . . 7 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴o (𝐵 ·o 𝑥)) = (𝐴o 𝑦𝑥 (𝐵 ·o 𝑦)))
55 limord 6243 . . . . . . . . . . . 12 (Lim 𝑥 → Ord 𝑥)
56 ordelon 6208 . . . . . . . . . . . 12 ((Ord 𝑥𝑦𝑥) → 𝑦 ∈ On)
5755, 56sylan 580 . . . . . . . . . . 11 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
5857, 33sylan2 592 . . . . . . . . . 10 ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐵 ·o 𝑦) ∈ On)
5958anassrs 468 . . . . . . . . 9 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑦𝑥) → (𝐵 ·o 𝑦) ∈ On)
6059ralrimiva 3179 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → ∀𝑦𝑥 (𝐵 ·o 𝑦) ∈ On)
61 0ellim 6246 . . . . . . . . . 10 (Lim 𝑥 → ∅ ∈ 𝑥)
6261ne0d 4298 . . . . . . . . 9 (Lim 𝑥𝑥 ≠ ∅)
6362adantl 482 . . . . . . . 8 ((𝐵 ∈ On ∧ Lim 𝑥) → 𝑥 ≠ ∅)
64 vex 3495 . . . . . . . . . 10 𝑤 ∈ V
65 oelim 8148 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6644, 65mpan2 687 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝑤 ∈ V ∧ Lim 𝑤)) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6717, 66mpan 686 . . . . . . . . . 10 ((𝑤 ∈ V ∧ Lim 𝑤) → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
6864, 67mpan 686 . . . . . . . . 9 (Lim 𝑤 → (𝐴o 𝑤) = 𝑧𝑤 (𝐴o 𝑧))
69 oewordi 8206 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7044, 69mpan2 687 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝐴 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
7117, 70mp3an3 1441 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤 → (𝐴o 𝑧) ⊆ (𝐴o 𝑤)))
72713impia 1109 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On ∧ 𝑧𝑤) → (𝐴o 𝑧) ⊆ (𝐴o 𝑤))
7368, 72onoviun 7969 . . . . . . . 8 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐵 ·o 𝑦) ∈ On ∧ 𝑥 ≠ ∅) → (𝐴o 𝑦𝑥 (𝐵 ·o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
7443, 60, 63, 73mp3an2i 1457 . . . . . . 7 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴o 𝑦𝑥 (𝐵 ·o 𝑦)) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
7554, 74eqtrd 2853 . . . . . 6 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴o (𝐵 ·o 𝑥)) = 𝑦𝑥 (𝐴o (𝐵 ·o 𝑦)))
7651, 75eqeq12d 2834 . . . . 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 247 . . . 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 7568 . 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 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492  wss 3933  c0 4288   ciun 4910  Ord word 6183  Oncon0 6184  Lim wlim 6185  suc csuc 6186  (class class class)co 7145  1oc1o 8084   +o coa 8088   ·o comu 8089  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-oexp 8097
This theorem is referenced by:  oeoe  8214
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