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Theorem oeoa 7722
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 7684 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∧ 𝐶 = ∅)))
21biimpar 501 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (𝐵 +𝑜 𝐶) = ∅)
32oveq2d 6706 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 ∅))
4 oveq2 6698 . . . . . . . . . 10 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
5 oveq2 6698 . . . . . . . . . . 11 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = (∅ ↑𝑜 ∅))
6 oe0m0 7645 . . . . . . . . . . 11 (∅ ↑𝑜 ∅) = 1𝑜
75, 6syl6eq 2701 . . . . . . . . . 10 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = 1𝑜)
84, 7oveqan12d 6709 . . . . . . . . 9 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 ∅) ·𝑜 1𝑜))
9 0elon 5816 . . . . . . . . . . 11 ∅ ∈ On
10 oecl 7662 . . . . . . . . . . 11 ((∅ ∈ On ∧ ∅ ∈ On) → (∅ ↑𝑜 ∅) ∈ On)
119, 9, 10mp2an 708 . . . . . . . . . 10 (∅ ↑𝑜 ∅) ∈ On
12 om1 7667 . . . . . . . . . 10 ((∅ ↑𝑜 ∅) ∈ On → ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅))
1311, 12ax-mp 5 . . . . . . . . 9 ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅)
148, 13syl6eq 2701 . . . . . . . 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 2688 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
17 oacl 7660 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +𝑜 𝐶) ∈ On)
18 on0eln0 5818 . . . . . . . . . 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 7646 . . . . . . . . . 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 2859 . . . . . . . . 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 501 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅)
25 on0eln0 5818 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2625adantr 480 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
27 on0eln0 5818 . . . . . . . . . . . 12 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2827adantl 481 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶𝐶 ≠ ∅))
2926, 28orbi12d 746 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅)))
30 neorian 2917 . . . . . . . . . 10 ((𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅))
3129, 30syl6bb 276 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
32 oe0m1 7646 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3332biimpa 500 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
3433oveq1d 6705 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ·𝑜 (∅ ↑𝑜 𝐶)))
35 oecl 7662 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 𝐶) ∈ On)
369, 35mpan 706 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (∅ ↑𝑜 𝐶) ∈ On)
37 om0r 7664 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐶) ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3934, 38sylan9eq 2705 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4039an32s 863 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
41 oe0m1 7646 . . . . . . . . . . . . . . 15 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
4241biimpa 500 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
4342oveq2d 6706 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
44 oecl 7662 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
459, 44mpan 706 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
46 om0 7642 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4745, 46syl 17 . . . . . . . . . . . . 13 (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4843, 47sylan9eqr 2707 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4948anassrs 681 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5040, 49jaodan 843 . . . . . . . . . 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 250 . . . . . . . 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 2688 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
5516, 54pm2.61dan 849 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
56 oveq1 6697 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 +𝑜 𝐶)))
57 oveq1 6697 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
58 oveq1 6697 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐶) = (∅ ↑𝑜 𝐶))
5957, 58oveq12d 6708 . . . . . 6 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
6056, 59eqeq12d 2666 . . . . 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 445 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
63 oveq1 6697 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)))
64 oveq1 6697 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
65 oveq1 6697 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))
6664, 65oveq12d 6708 . . . . . . . 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 2666 . . . . . . 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 6697 . . . . . . . . 9 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (𝐵 +𝑜 𝐶) = (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶))
7069oveq2d 6706 . . . . . . . 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 6698 . . . . . . . . 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 6705 . . . . . . . 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 2666 . . . . . . 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 2718 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
76 eleq2 2719 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
7775, 76anbi12d 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))))
78 eleq1 2718 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
79 eleq2 2719 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
8078, 79anbi12d 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))))
81 1on 7612 . . . . . . . . . 10 1𝑜 ∈ On
82 0lt1o 7629 . . . . . . . . . 10 ∅ ∈ 1𝑜
8381, 82pm3.2i 470 . . . . . . . . 9 (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
8477, 80, 83elimhyp 4179 . . . . . . . 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 4183 . . . . . . 7 if(𝐵 ∈ On, 𝐵, 1𝑜) ∈ On
8885, 86, 87oeoalem 7721 . . . . . 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 4173 . . . . 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 863 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
9262, 91oe0lem 7638 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
93923impb 1279 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 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 coa 7602   ·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:  oeoelem  7723  infxpenc  8879
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