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Theorem oeoe 7884
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 6850 . . . . . . . . . . . 12 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7805 . . . . . . . . . . . 12 (∅ ↑𝑜 ∅) = 1𝑜
31, 2syl6eq 2815 . . . . . . . . . . 11 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = 1𝑜)
43oveq1d 6857 . . . . . . . . . 10 (𝐵 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (1𝑜𝑜 𝐶))
5 oe1m 7830 . . . . . . . . . 10 (𝐶 ∈ On → (1𝑜𝑜 𝐶) = 1𝑜)
64, 5sylan9eqr 2821 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
76adantll 705 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
8 oveq2 6850 . . . . . . . . . 10 (𝐶 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 ∅))
9 0elon 5961 . . . . . . . . . . . 12 ∅ ∈ On
10 oecl 7822 . . . . . . . . . . . 12 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
119, 10mpan 681 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
12 oe0 7807 . . . . . . . . . . 11 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
1311, 12syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
148, 13sylan9eqr 2821 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
1514adantlr 706 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
167, 15jaodan 980 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
17 om00 7860 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
1817biimpar 469 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·𝑜 𝐶) = ∅)
1918oveq2d 6858 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 ∅))
2019, 2syl6eq 2815 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = 1𝑜)
2116, 20eqtr4d 2802 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
22 on0eln0 5963 . . . . . . . . . 10 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
23 on0eln0 5963 . . . . . . . . . 10 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2422, 23bi2anan9 629 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25 neanior 3029 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
2624, 25syl6bb 278 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27 oe0m1 7806 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
2827biimpa 468 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
2928oveq1d 6857 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 𝐶))
30 oe0m1 7806 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
3130biimpa 468 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
3229, 31sylan9eq 2819 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
3332an4s 650 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
34 om00el 7861 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·𝑜 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35 omcl 7821 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·𝑜 𝐶) ∈ On)
36 oe0m1 7806 . . . . . . . . . . . . 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 272 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅))
3938biimpa 468 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅)
4033, 39eqtr4d 2802 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4140ex 401 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
4226, 41sylbird 251 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∨ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
4342imp 395 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4421, 43pm2.61dan 847 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
45 oveq1 6849 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
4645oveq1d 6857 . . . . . 6 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶))
47 oveq1 6849 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4846, 47eqeq12d 2780 . . . . 5 (𝐴 = ∅ → (((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)) ↔ ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶))))
4944, 48syl5ibr 237 . . . 4 (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶))))
5049impcom 396 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
51 oveq1 6849 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
5251oveq1d 6857 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ↑𝑜 𝐶))
53 oveq1 6849 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 (𝐵 ·𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 ·𝑜 𝐶)))
5452, 53eqeq12d 2780 . . . . . . 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 331 . . . . . 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 2832 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
57 eleq2 2833 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
5856, 57anbi12d 624 . . . . . . . . 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 2832 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
60 eleq2 2833 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
6159, 60anbi12d 624 . . . . . . . . 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 7771 . . . . . . . . . 10 1𝑜 ∈ On
63 0lt1o 7789 . . . . . . . . . 10 ∅ ∈ 1𝑜
6462, 63pm3.2i 462 . . . . . . . . 9 (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
6558, 61, 64elimhyp 4306 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
6665simpli 476 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
6765simpri 479 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
6866, 67oeoelem 7883 . . . . . 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 4299 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶))))
7069imp 395 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
7170an32s 642 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
7250, 71oe0lem 7798 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
73723impb 1143 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  c0 4079  ifcif 4243  Oncon0 5908  (class class class)co 6842  1𝑜c1o 7757   ·𝑜 comu 7762  𝑜 coe 7763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-omul 7769  df-oexp 7770
This theorem is referenced by:  infxpenc  9092
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