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Theorem oeoe 8637
Description: Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.)
Assertion
Ref Expression
oeoe ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))

Proof of Theorem oeoe
StepHypRef Expression
1 oveq2 7439 . . . . . . . . . . . 12 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
2 oe0m0 8558 . . . . . . . . . . . 12 (∅ ↑o ∅) = 1o
31, 2eqtrdi 2793 . . . . . . . . . . 11 (𝐵 = ∅ → (∅ ↑o 𝐵) = 1o)
43oveq1d 7446 . . . . . . . . . 10 (𝐵 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = (1oo 𝐶))
5 oe1m 8583 . . . . . . . . . 10 (𝐶 ∈ On → (1oo 𝐶) = 1o)
64, 5sylan9eqr 2799 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
76adantll 714 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
8 oveq2 7439 . . . . . . . . . 10 (𝐶 = ∅ → ((∅ ↑o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o ∅))
9 0elon 6438 . . . . . . . . . . . 12 ∅ ∈ On
10 oecl 8575 . . . . . . . . . . . 12 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
119, 10mpan 690 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
12 oe0 8560 . . . . . . . . . . 11 ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
1311, 12syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ((∅ ↑o 𝐵) ↑o ∅) = 1o)
148, 13sylan9eqr 2799 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
1514adantlr 715 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
167, 15jaodan 960 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = 1o)
17 om00 8613 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·o 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
1817biimpar 477 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·o 𝐶) = ∅)
1918oveq2d 7447 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = (∅ ↑o ∅))
2019, 2eqtrdi 2793 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑o (𝐵 ·o 𝐶)) = 1o)
2116, 20eqtr4d 2780 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
22 on0eln0 6440 . . . . . . . . . 10 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
23 on0eln0 6440 . . . . . . . . . 10 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2422, 23bi2anan9 638 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25 neanior 3035 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
2624, 25bitrdi 287 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27 oe0m1 8559 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
2827biimpa 476 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
2928oveq1d 7446 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o 𝐶))
30 oe0m1 8559 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑o 𝐶) = ∅))
3130biimpa 476 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑o 𝐶) = ∅)
3229, 31sylan9eq 2797 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
3332an4s 660 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = ∅)
34 om00el 8614 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35 omcl 8574 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·o 𝐶) ∈ On)
36 oe0m1 8559 . . . . . . . . . . . . 13 ((𝐵 ·o 𝐶) ∈ On → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ↑o (𝐵 ·o 𝐶)) = ∅))
3735, 36syl 17 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·o 𝐶) ↔ (∅ ↑o (𝐵 ·o 𝐶)) = ∅))
3834, 37bitr3d 281 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (∅ ↑o (𝐵 ·o 𝐶)) = ∅))
3938biimpa 476 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑o (𝐵 ·o 𝐶)) = ∅)
4033, 39eqtr4d 2780 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
4140ex 412 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶))))
4226, 41sylbird 260 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∨ 𝐶 = ∅) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶))))
4342imp 406 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
4421, 43pm2.61dan 813 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶)))
45 oveq1 7438 . . . . . . 7 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
4645oveq1d 7446 . . . . . 6 (𝐴 = ∅ → ((𝐴o 𝐵) ↑o 𝐶) = ((∅ ↑o 𝐵) ↑o 𝐶))
47 oveq1 7438 . . . . . 6 (𝐴 = ∅ → (𝐴o (𝐵 ·o 𝐶)) = (∅ ↑o (𝐵 ·o 𝐶)))
4846, 47eqeq12d 2753 . . . . 5 (𝐴 = ∅ → (((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)) ↔ ((∅ ↑o 𝐵) ↑o 𝐶) = (∅ ↑o (𝐵 ·o 𝐶))))
4944, 48imbitrrid 246 . . . 4 (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶))))
5049impcom 407 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))
51 oveq1 7438 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴o 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵))
5251oveq1d 7446 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → ((𝐴o 𝐵) ↑o 𝐶) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶))
53 oveq1 7438 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴o (𝐵 ·o 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶)))
5452, 53eqeq12d 2753 . . . . . . 7 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)) ↔ ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶))))
5554imbi2d 340 . . . . . 6 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶))) ↔ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶)))))
56 eleq1 2829 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
57 eleq2 2830 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
5856, 57anbi12d 632 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o))))
59 eleq1 2829 . . . . . . . . . 10 (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (1o ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
60 eleq2 2830 . . . . . . . . . 10 (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 1o ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
6159, 60anbi12d 632 . . . . . . . . 9 (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → ((1o ∈ On ∧ ∅ ∈ 1o) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o))))
62 1on 8518 . . . . . . . . . 10 1o ∈ On
63 0lt1o 8542 . . . . . . . . . 10 ∅ ∈ 1o
6462, 63pm3.2i 470 . . . . . . . . 9 (1o ∈ On ∧ ∅ ∈ 1o)
6558, 61, 64elimhyp 4591 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o))
6665simpli 483 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On
6765simpri 485 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)
6866, 67oeoelem 8636 . . . . . 6 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵) ↑o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 ·o 𝐶)))
6955, 68dedth 4584 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶))))
7069imp 406 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))
7170an32s 652 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))
7250, 71oe0lem 8551 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))
73723impb 1115 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  c0 4333  ifcif 4525  Oncon0 6384  (class class class)co 7431  1oc1o 8499   ·o comu 8504  o coe 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by:  infxpenc  10058
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