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Theorem oeoe 7724
 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 6698 . . . . . . . . . . . 12 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7645 . . . . . . . . . . . 12 (∅ ↑𝑜 ∅) = 1𝑜
31, 2syl6eq 2701 . . . . . . . . . . 11 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = 1𝑜)
43oveq1d 6705 . . . . . . . . . 10 (𝐵 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (1𝑜𝑜 𝐶))
5 oe1m 7670 . . . . . . . . . 10 (𝐶 ∈ On → (1𝑜𝑜 𝐶) = 1𝑜)
64, 5sylan9eqr 2707 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
76adantll 750 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
8 oveq2 6698 . . . . . . . . . 10 (𝐶 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 ∅))
9 0elon 5816 . . . . . . . . . . . 12 ∅ ∈ On
10 oecl 7662 . . . . . . . . . . . 12 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
119, 10mpan 706 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
12 oe0 7647 . . . . . . . . . . 11 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
1311, 12syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
148, 13sylan9eqr 2707 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
1514adantlr 751 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
167, 15jaodan 843 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
17 om00 7700 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
1817biimpar 501 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·𝑜 𝐶) = ∅)
1918oveq2d 6706 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 ∅))
2019, 2syl6eq 2701 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = 1𝑜)
2116, 20eqtr4d 2688 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
22 on0eln0 5818 . . . . . . . . . 10 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
23 on0eln0 5818 . . . . . . . . . 10 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2422, 23bi2anan9 935 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25 neanior 2915 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
2624, 25syl6bb 276 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27 oe0m1 7646 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
2827biimpa 500 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
2928oveq1d 6705 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 𝐶))
30 oe0m1 7646 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
3130biimpa 500 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
3229, 31sylan9eq 2705 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
3332an4s 886 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
34 om00el 7701 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·𝑜 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35 omcl 7661 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·𝑜 𝐶) ∈ On)
36 oe0m1 7646 . . . . . . . . . . . . 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 270 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅))
3938biimpa 500 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅)
4033, 39eqtr4d 2688 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4140ex 449 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
4226, 41sylbird 250 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∨ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
4342imp 444 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4421, 43pm2.61dan 849 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
45 oveq1 6697 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
4645oveq1d 6705 . . . . . 6 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶))
47 oveq1 6697 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4846, 47eqeq12d 2666 . . . . 5 (𝐴 = ∅ → (((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)) ↔ ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
4944, 48syl5ibr 236 . . . 4 (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶))))
5049impcom 445 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
51 oveq1 6697 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
5251oveq1d 6705 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ↑𝑜 𝐶))
53 oveq1 6697 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 (𝐵 ·𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 ·𝑜 𝐶)))
5452, 53eqeq12d 2666 . . . . . . 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 329 . . . . . 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 2718 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
57 eleq2 2719 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
5856, 57anbi12d 747 . . . . . . . . 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 2718 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
60 eleq2 2719 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
6159, 60anbi12d 747 . . . . . . . . 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 7612 . . . . . . . . . 10 1𝑜 ∈ On
63 0lt1o 7629 . . . . . . . . . 10 ∅ ∈ 1𝑜
6462, 63pm3.2i 470 . . . . . . . . 9 (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
6558, 61, 64elimhyp 4179 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
6665simpli 473 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
6765simpri 477 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
6866, 67oeoelem 7723 . . . . . 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 4172 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶))))
7069imp 444 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
7170an32s 863 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
7250, 71oe0lem 7638 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
73723impb 1279 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 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∅c0 3948  ifcif 4119  Oncon0 5761  (class class class)co 6690  1𝑜c1o 7598   ·𝑜 comu 7603   ↑𝑜 coe 7604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611 This theorem is referenced by:  infxpenc  8879
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