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Theorem oeoa 7914
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 7876 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) = ∅ ↔ (𝐵 = ∅ ∧ 𝐶 = ∅)))
21biimpar 465 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (𝐵 +𝑜 𝐶) = ∅)
32oveq2d 6890 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 ∅))
4 oveq2 6882 . . . . . . . . . 10 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
5 oveq2 6882 . . . . . . . . . . 11 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = (∅ ↑𝑜 ∅))
6 oe0m0 7837 . . . . . . . . . . 11 (∅ ↑𝑜 ∅) = 1𝑜
75, 6syl6eq 2856 . . . . . . . . . 10 (𝐶 = ∅ → (∅ ↑𝑜 𝐶) = 1𝑜)
84, 7oveqan12d 6893 . . . . . . . . 9 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 ∅) ·𝑜 1𝑜))
9 0elon 5991 . . . . . . . . . . 11 ∅ ∈ On
10 oecl 7854 . . . . . . . . . . 11 ((∅ ∈ On ∧ ∅ ∈ On) → (∅ ↑𝑜 ∅) ∈ On)
119, 9, 10mp2an 675 . . . . . . . . . 10 (∅ ↑𝑜 ∅) ∈ On
12 om1 7859 . . . . . . . . . 10 ((∅ ↑𝑜 ∅) ∈ On → ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅))
1311, 12ax-mp 5 . . . . . . . . 9 ((∅ ↑𝑜 ∅) ·𝑜 1𝑜) = (∅ ↑𝑜 ∅)
148, 13syl6eq 2856 . . . . . . . 8 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
1514adantl 469 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ↑𝑜 ∅))
163, 15eqtr4d 2843 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
17 oacl 7852 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +𝑜 𝐶) ∈ On)
18 on0eln0 5993 . . . . . . . . . 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 7838 . . . . . . . . . 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 3014 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +𝑜 𝐶) ≠ ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2319, 21, 223bitr3d 300 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2423biimpar 465 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ∅)
25 on0eln0 5993 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2625adantr 468 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
27 on0eln0 5993 . . . . . . . . . . . 12 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2827adantl 469 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶𝐶 ≠ ∅))
2926, 28orbi12d 933 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅)))
30 neorian 3072 . . . . . . . . . 10 ((𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅))
3129, 30syl6bb 278 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
32 oe0m1 7838 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3332biimpa 464 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
3433oveq1d 6889 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = (∅ ·𝑜 (∅ ↑𝑜 𝐶)))
35 oecl 7854 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 𝐶) ∈ On)
369, 35mpan 673 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (∅ ↑𝑜 𝐶) ∈ On)
37 om0r 7856 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐶) ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
3934, 38sylan9eq 2860 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐶 ∈ On) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4039an32s 634 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
41 oe0m1 7838 . . . . . . . . . . . . . . 15 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑𝑜 𝐶) = ∅))
4241biimpa 464 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑𝑜 𝐶) = ∅)
4342oveq2d 6890 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
44 oecl 7854 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑𝑜 𝐵) ∈ On)
459, 44mpan 673 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) ∈ On)
46 om0 7834 . . . . . . . . . . . . . 14 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4745, 46syl 17 . . . . . . . . . . . . 13 (𝐵 ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
4843, 47sylan9eqr 2862 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
4948anassrs 455 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5040, 49jaodan 971 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5150ex 399 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
5231, 51sylbird 251 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅))
5352imp 395 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)) = ∅)
5424, 53eqtr4d 2843 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
5516, 54pm2.61dan 838 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
56 oveq1 6881 . . . . . 6 (𝐴 = ∅ → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (∅ ↑𝑜 (𝐵 +𝑜 𝐶)))
57 oveq1 6881 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
58 oveq1 6881 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐶) = (∅ ↑𝑜 𝐶))
5957, 58oveq12d 6892 . . . . . 6 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶)))
6056, 59eqeq12d 2821 . . . . 5 (𝐴 = ∅ → ((𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)) ↔ (∅ ↑𝑜 (𝐵 +𝑜 𝐶)) = ((∅ ↑𝑜 𝐵) ·𝑜 (∅ ↑𝑜 𝐶))))
6155, 60syl5ibr 237 . . . 4 (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))))
6261impcom 396 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
63 oveq1 6881 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 (𝐵 +𝑜 𝐶)))
64 oveq1 6881 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐵))
65 oveq1 6881 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴𝑜 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ↑𝑜 𝐶))
6664, 65oveq12d 6892 . . . . . . . 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 2821 . . . . . . 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 331 . . . . . 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 6881 . . . . . . . . 9 (𝐵 = if(𝐵 ∈ On, 𝐵, 1𝑜) → (𝐵 +𝑜 𝐶) = (if(𝐵 ∈ On, 𝐵, 1𝑜) +𝑜 𝐶))
7069oveq2d 6890 . . . . . . . 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 6882 . . . . . . . . 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 6889 . . . . . . . 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 2821 . . . . . . 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 331 . . . . . 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 2873 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
76 eleq2 2874 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
7775, 76anbi12d 618 . . . . . . . . 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 2873 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (1𝑜 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On))
79 eleq2 2874 . . . . . . . . . 10 (1𝑜 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) → (∅ ∈ 1𝑜 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)))
8078, 79anbi12d 618 . . . . . . . . 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 7803 . . . . . . . . . 10 1𝑜 ∈ On
82 0lt1o 7821 . . . . . . . . . 10 ∅ ∈ 1𝑜
8381, 82pm3.2i 458 . . . . . . . . 9 (1𝑜 ∈ On ∧ ∅ ∈ 1𝑜)
8477, 80, 83elimhyp 4342 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜))
8584simpli 472 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜) ∈ On
8684simpri 475 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1𝑜)
8781elimel 4346 . . . . . . 7 if(𝐵 ∈ On, 𝐵, 1𝑜) ∈ On
8885, 86, 87oeoalem 7913 . . . . . 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 4336 . . . . 5 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶))))
9089impr 444 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
9190an32s 634 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
9262, 91oe0lem 7830 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
93923impb 1136 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 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156  wne 2978  c0 4116  ifcif 4279  Oncon0 5936  (class class class)co 6874  1𝑜c1o 7789   +𝑜 coa 7793   ·𝑜 comu 7794  𝑜 coe 7795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-omul 7801  df-oexp 7802
This theorem is referenced by:  oeoelem  7915  infxpenc  9124
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