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Theorem oeoe 7624
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 6612 . . . . . . . . . . . 12 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7545 . . . . . . . . . . . 12 (∅ ↑𝑜 ∅) = 1𝑜
31, 2syl6eq 2671 . . . . . . . . . . 11 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = 1𝑜)
43oveq1d 6619 . . . . . . . . . 10 (𝐵 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (1𝑜𝑜 𝐶))
5 oe1m 7570 . . . . . . . . . 10 (𝐶 ∈ On → (1𝑜𝑜 𝐶) = 1𝑜)
64, 5sylan9eqr 2677 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
76adantll 749 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
8 oveq2 6612 . . . . . . . . . 10 (𝐶 = ∅ → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 ∅))
9 0elon 5737 . . . . . . . . . . . 12 ∅ ∈ On
10 oecl 7562 . . . . . . . . . . . 12 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
119, 10mpan 705 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
12 oe0 7547 . . . . . . . . . . 11 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
1311, 12syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ↑𝑜 ∅) = 1𝑜)
148, 13sylan9eqr 2677 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
1514adantlr 750 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
167, 15jaodan 825 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = 1𝑜)
17 om00 7600 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ·𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∨ 𝐶 = ∅)))
1817biimpar 502 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (𝐵 ·𝑜 𝐶) = ∅)
1918oveq2d 6620 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 ∅))
2019, 2syl6eq 2671 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = 1𝑜)
2116, 20eqtr4d 2658 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
22 on0eln0 5739 . . . . . . . . . 10 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
23 on0eln0 5739 . . . . . . . . . 10 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2422, 23bi2anan9 916 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅)))
25 neanior 2882 . . . . . . . . 9 ((𝐵 ≠ ∅ ∧ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅))
2624, 25syl6bb 276 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)))
27 oe0m1 7546 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
2827biimpa 501 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
2928oveq1d 6619 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 𝐶))
30 oe0m1 7546 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
3130biimpa 501 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
3229, 31sylan9eq 2675 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
3332an4s 868 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = ∅)
34 om00el 7601 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ·𝑜 𝐶) ↔ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
35 omcl 7561 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ·𝑜 𝐶) ∈ On)
36 oe0m1 7546 . . . . . . . . . . . . 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 501 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)) = ∅)
4033, 39eqtr4d 2658 . . . . . . . . 9 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4140ex 450 . . . . . . . 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 445 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∨ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4421, 43pm2.61dan 831 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
45 oveq1 6611 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
4645oveq1d 6619 . . . . . 6 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = ((∅ ↑𝑜 𝐵) ↑𝑜 𝐶))
47 oveq1 6611 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 (𝐵 ·𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 ·𝑜 𝐶)))
4846, 47eqeq12d 2636 . . . . 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 446 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
51 oveq1 6611 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
5251oveq1d 6619 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ↑𝑜 𝐶))
53 oveq1 6611 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 (𝐵 ·𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 ·𝑜 𝐶)))
5452, 53eqeq12d 2636 . . . . . . 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 330 . . . . . 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 2686 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
57 eleq2 2687 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
5856, 57anbi12d 746 . . . . . . . . 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 2686 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
60 eleq2 2687 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
6159, 60anbi12d 746 . . . . . . . . 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 7512 . . . . . . . . . 10 1𝑜 ∈ On
63 0lt1o 7529 . . . . . . . . . 10 ∅ ∈ 1𝑜
6462, 63pm3.2i 471 . . . . . . . . 9 (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
6558, 61, 64elimhyp 4118 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
6665simpli 474 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
6765simpri 478 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
6866, 67oeoelem 7623 . . . . . 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 4111 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶))))
7069imp 445 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
7170an32s 845 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
7250, 71oe0lem 7538 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
73723impb 1257 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  c0 3891  ifcif 4058  Oncon0 5682  (class class class)co 6604  1𝑜c1o 7498   ·𝑜 comu 7503  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511
This theorem is referenced by:  infxpenc  8785
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