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Theorem oeoa 7622
Description: Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.)
Assertion
Ref Expression
oeoa ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))

Proof of Theorem oeoa
StepHypRef Expression
1 oa00 7584 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∧ 𝐶 = ∅)))
21biimpar 502 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (𝐵 +𝑜 𝐶) = ∅)
32oveq2d 6620 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 ∅))
4 oveq2 6612 . . . . . . . . . 10 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
5 oveq2 6612 . . . . . . . . . . 11 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = (∅ ↑𝑜 ∅))
6 oe0m0 7545 . . . . . . . . . . 11 (∅ ↑𝑜 ∅) = 1𝑜
75, 6syl6eq 2671 . . . . . . . . . 10 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = 1𝑜)
84, 7oveqan12d 6623 . . . . . . . . 9 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 ∅) ·𝑜 1𝑜))
9 0elon 5737 . . . . . . . . . . 11 ∅ ∈ On
10 oecl 7562 . . . . . . . . . . 11 ((∅ ∈ On ∧ ∅ ∈ On) → (∅ ↑𝑜 ∅) ∈ On)
119, 9, 10mp2an 707 . . . . . . . . . 10 (∅ ↑𝑜 ∅) ∈ On
12 om1 7567 . . . . . . . . . 10 ((∅ ↑𝑜 ∅) ∈ On → ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅))
1311, 12ax-mp 5 . . . . . . . . 9 ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅)
148, 13syl6eq 2671 . . . . . . . 8 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
1514adantl 482 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
163, 15eqtr4d 2658 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
17 oacl 7560 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +𝑜 𝐶) ∈ On)
18 on0eln0 5739 . . . . . . . . . 10 ((𝐵 +𝑜 𝐶) ∈ On → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (𝐵 +𝑜 𝐶) ≠ ∅))
1917, 18syl 17 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (𝐵 +𝑜 𝐶) ≠ ∅))
20 oe0m1 7546 . . . . . . . . . 10 ((𝐵 +𝑜 𝐶) ∈ On → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅))
2117, 20syl 17 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 +𝑜 𝐶) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅))
221necon3abid 2826 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) ≠ ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2319, 21, 223bitr3d 298 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2423biimpar 502 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅)
25 on0eln0 5739 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2625adantr 481 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
27 on0eln0 5739 . . . . . . . . . . . 12 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2827adantl 482 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶𝐶 ≠ ∅))
2926, 28orbi12d 745 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅)))
30 neorian 2884 . . . . . . . . . 10 ((𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅))
3129, 30syl6bb 276 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
32 oe0m1 7546 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3332biimpa 501 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
3433oveq1d 6619 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ·𝑜 (∅ ↑𝑜 𝐶)))
35 oecl 7562 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 𝐶) ∈ On)
369, 35mpan 705 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (∅ ↑𝑜 𝐶) ∈ On)
37 om0r 7564 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐶) ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3934, 38sylan9eq 2675 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4039an32s 845 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
41 oe0m1 7546 . . . . . . . . . . . . . . 15 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
4241biimpa 501 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
4342oveq2d 6620 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
44 oecl 7562 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
459, 44mpan 705 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
46 om0 7542 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4745, 46syl 17 . . . . . . . . . . . . 13 (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4843, 47sylan9eqr 2677 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4948anassrs 679 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5040, 49jaodan 825 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5150ex 450 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
5231, 51sylbird 250 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
5352imp 445 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5424, 53eqtr4d 2658 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
5516, 54pm2.61dan 831 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
56 oveq1 6611 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 +𝑜 𝐶)))
57 oveq1 6611 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
58 oveq1 6611 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐶) = (∅ ↑𝑜 𝐶))
5957, 58oveq12d 6622 . . . . . 6 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
6056, 59eqeq12d 2636 . . . . 5 (𝐴 = ∅ → ((𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶))))
6155, 60syl5ibr 236 . . . 4 (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))))
6261impcom 446 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
63 oveq1 6611 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)))
64 oveq1 6611 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
65 oveq1 6611 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))
6664, 65oveq12d 6622 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))
6763, 66eqeq12d 2636 . . . . . . 7 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))))
6867imbi2d 330 . . . . . 6 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → ((𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))) ↔ (𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))))
69 oveq1 6611 . . . . . . . . 9 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (𝐵 +𝑜 𝐶) = (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶))
7069oveq2d 6620 . . . . . . . 8 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)))
71 oveq2 6612 . . . . . . . . 9 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)))
7271oveq1d 6619 . . . . . . . 8 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))
7370, 72eqeq12d 2636 . . . . . . 7 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)) ↔ (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))))
7473imbi2d 330 . . . . . 6 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → ((𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))) ↔ (𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))))
75 eleq1 2686 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
76 eleq2 2687 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
7775, 76anbi12d 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))))
78 eleq1 2686 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
79 eleq2 2687 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
8078, 79anbi12d 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))))
81 1on 7512 . . . . . . . . . 10 1𝑜 ∈ On
82 0lt1o 7529 . . . . . . . . . 10 ∅ ∈ 1𝑜
8381, 82pm3.2i 471 . . . . . . . . 9 (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
8477, 80, 83elimhyp 4118 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
8584simpli 474 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
8684simpri 478 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
8781elimel 4122 . . . . . . 7 if(𝐵 ∈ On, 𝐵, 1𝑜) ∈ On
8885, 86, 87oeoalem 7621 . . . . . 6 (𝐶 ∈ On → (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶)) = ((if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 if(𝐵 ∈ On, 𝐵, 1𝑜)) ·𝑜 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶)))
8968, 74, 88dedth2h 4112 . . . . 5 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))))
9089impr 648 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
9190an32s 845 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
9262, 91oe0lem 7538 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
93923impb 1257 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 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 coa 7502   ·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:  oeoelem  7623  infxpenc  8785
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