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Theorem oeoa 7564
 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 7526 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∧ 𝐶 = ∅)))
21biimpar 501 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (𝐵 +𝑜 𝐶) = ∅)
32oveq2d 6565 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 ∅))
4 oveq2 6557 . . . . . . . . . 10 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
5 oveq2 6557 . . . . . . . . . . 11 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = (∅ ↑𝑜 ∅))
6 oe0m0 7487 . . . . . . . . . . 11 (∅ ↑𝑜 ∅) = 1𝑜
75, 6syl6eq 2660 . . . . . . . . . 10 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = 1𝑜)
84, 7oveqan12d 6568 . . . . . . . . 9 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 ∅) ·𝑜 1𝑜))
9 0elon 5695 . . . . . . . . . . 11 ∅ ∈ On
10 oecl 7504 . . . . . . . . . . 11 ((∅ ∈ On ∧ ∅ ∈ On) → (∅ ↑𝑜 ∅) ∈ On)
119, 9, 10mp2an 704 . . . . . . . . . 10 (∅ ↑𝑜 ∅) ∈ On
12 om1 7509 . . . . . . . . . 10 ((∅ ↑𝑜 ∅) ∈ On → ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅))
1311, 12ax-mp 5 . . . . . . . . 9 ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅)
148, 13syl6eq 2660 . . . . . . . 8 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
1514adantl 481 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
163, 15eqtr4d 2647 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
17 oacl 7502 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +𝑜 𝐶) ∈ On)
18 on0eln0 5697 . . . . . . . . . 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 7488 . . . . . . . . . 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 2818 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) ≠ ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2319, 21, 223bitr3d 297 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2423biimpar 501 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅)
25 on0eln0 5697 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2625adantr 480 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
27 on0eln0 5697 . . . . . . . . . . . 12 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2827adantl 481 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶𝐶 ≠ ∅))
2926, 28orbi12d 742 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅)))
30 neorian 2876 . . . . . . . . . 10 ((𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅))
3129, 30syl6bb 275 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
32 oe0m1 7488 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3332biimpa 500 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
3433oveq1d 6564 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ·𝑜 (∅ ↑𝑜 𝐶)))
35 oecl 7504 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 𝐶) ∈ On)
369, 35mpan 702 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (∅ ↑𝑜 𝐶) ∈ On)
37 om0r 7506 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐶) ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3934, 38sylan9eq 2664 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4039an32s 842 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
41 oe0m1 7488 . . . . . . . . . . . . . . 15 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
4241biimpa 500 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
4342oveq2d 6565 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
44 oecl 7504 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
459, 44mpan 702 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
46 om0 7484 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4745, 46syl 17 . . . . . . . . . . . . 13 (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4843, 47sylan9eqr 2666 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4948anassrs 678 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5040, 49jaodan 822 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5150ex 449 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
5231, 51sylbird 249 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
5352imp 444 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5424, 53eqtr4d 2647 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
5516, 54pm2.61dan 828 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
56 oveq1 6556 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 +𝑜 𝐶)))
57 oveq1 6556 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
58 oveq1 6556 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐶) = (∅ ↑𝑜 𝐶))
5957, 58oveq12d 6567 . . . . . 6 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
6056, 59eqeq12d 2625 . . . . 5 (𝐴 = ∅ → ((𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶))))
6155, 60syl5ibr 235 . . . 4 (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))))
6261impcom 445 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
63 oveq1 6556 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)))
64 oveq1 6556 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
65 oveq1 6556 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))
6664, 65oveq12d 6567 . . . . . . . 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 2625 . . . . . . 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 329 . . . . . 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 6556 . . . . . . . . 9 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (𝐵 +𝑜 𝐶) = (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶))
7069oveq2d 6565 . . . . . . . 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 6557 . . . . . . . . 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 6564 . . . . . . . 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 2625 . . . . . . 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 329 . . . . . 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 2676 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
76 eleq2 2677 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
7775, 76anbi12d 743 . . . . . . . . 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 2676 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
79 eleq2 2677 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
8078, 79anbi12d 743 . . . . . . . . 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 7454 . . . . . . . . . 10 1𝑜 ∈ On
82 0lt1o 7471 . . . . . . . . . 10 ∅ ∈ 1𝑜
8381, 82pm3.2i 470 . . . . . . . . 9 (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
8477, 80, 83elimhyp 4096 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
8584simpli 473 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
8684simpri 477 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
8781elimel 4100 . . . . . . 7 if(𝐵 ∈ On, 𝐵, 1𝑜) ∈ On
8885, 86, 87oeoalem 7563 . . . . . 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 4090 . . . . 5 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))))
9089impr 647 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
9190an32s 842 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
9262, 91oe0lem 7480 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
93923impb 1252 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ifcif 4036  Oncon0 5640  (class class class)co 6549  1𝑜c1o 7440   +𝑜 coa 7444   ·𝑜 comu 7445   ↑𝑜 coe 7446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453 This theorem is referenced by:  oeoelem  7565  infxpenc  8724
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