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Theorem oeoa 8209
 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 8171 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +o 𝐶) = ∅ ↔ (𝐵 = ∅ ∧ 𝐶 = ∅)))
21biimpar 481 . . . . . . . 8 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (𝐵 +o 𝐶) = ∅)
32oveq2d 7152 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = (∅ ↑o ∅))
4 oveq2 7144 . . . . . . . . . 10 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
5 oveq2 7144 . . . . . . . . . . 11 (𝐶 = ∅ → (∅ ↑o 𝐶) = (∅ ↑o ∅))
6 oe0m0 8131 . . . . . . . . . . 11 (∅ ↑o ∅) = 1o
75, 6eqtrdi 2849 . . . . . . . . . 10 (𝐶 = ∅ → (∅ ↑o 𝐶) = 1o)
84, 7oveqan12d 7155 . . . . . . . . 9 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ((∅ ↑o ∅) ·o 1o))
9 0elon 6213 . . . . . . . . . . 11 ∅ ∈ On
10 oecl 8148 . . . . . . . . . . 11 ((∅ ∈ On ∧ ∅ ∈ On) → (∅ ↑o ∅) ∈ On)
119, 9, 10mp2an 691 . . . . . . . . . 10 (∅ ↑o ∅) ∈ On
12 om1 8154 . . . . . . . . . 10 ((∅ ↑o ∅) ∈ On → ((∅ ↑o ∅) ·o 1o) = (∅ ↑o ∅))
1311, 12ax-mp 5 . . . . . . . . 9 ((∅ ↑o ∅) ·o 1o) = (∅ ↑o ∅)
148, 13eqtrdi 2849 . . . . . . . 8 ((𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = (∅ ↑o ∅))
1514adantl 485 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = (∅ ↑o ∅))
163, 15eqtr4d 2836 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
17 oacl 8146 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +o 𝐶) ∈ On)
18 on0eln0 6215 . . . . . . . . . 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 8132 . . . . . . . . . 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 3023 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 +o 𝐶) ≠ ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2319, 21, 223bitr3d 312 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ↑o (𝐵 +o 𝐶)) = ∅ ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
2423biimpar 481 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = ∅)
25 on0eln0 6215 . . . . . . . . . . . 12 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2625adantr 484 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
27 on0eln0 6215 . . . . . . . . . . . 12 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
2827adantl 485 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶𝐶 ≠ ∅))
2926, 28orbi12d 916 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ (𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅)))
30 neorian 3081 . . . . . . . . . 10 ((𝐵 ≠ ∅ ∨ 𝐶 ≠ ∅) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅))
3129, 30syl6bb 290 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) ↔ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)))
32 oe0m1 8132 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
3332biimpa 480 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
3433oveq1d 7151 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = (∅ ·o (∅ ↑o 𝐶)))
35 oecl 8148 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐶 ∈ On) → (∅ ↑o 𝐶) ∈ On)
369, 35mpan 689 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (∅ ↑o 𝐶) ∈ On)
37 om0r 8150 . . . . . . . . . . . . . 14 ((∅ ↑o 𝐶) ∈ On → (∅ ·o (∅ ↑o 𝐶)) = ∅)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝐶 ∈ On → (∅ ·o (∅ ↑o 𝐶)) = ∅)
3934, 38sylan9eq 2853 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐶 ∈ On) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
4039an32s 651 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
41 oe0m1 8132 . . . . . . . . . . . . . . 15 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (∅ ↑o 𝐶) = ∅))
4241biimpa 480 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → (∅ ↑o 𝐶) = ∅)
4342oveq2d 7152 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ((∅ ↑o 𝐵) ·o ∅))
44 oecl 8148 . . . . . . . . . . . . . . 15 ((∅ ∈ On ∧ 𝐵 ∈ On) → (∅ ↑o 𝐵) ∈ On)
459, 44mpan 689 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ↑o 𝐵) ∈ On)
46 om0 8128 . . . . . . . . . . . . . 14 ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
4745, 46syl 17 . . . . . . . . . . . . 13 (𝐵 ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
4843, 47sylan9eqr 2855 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
4948anassrs 471 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
5040, 49jaodan 955 . . . . . . . . . 10 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
5150ex 416 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∅ ∈ 𝐵 ∨ ∅ ∈ 𝐶) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅))
5231, 51sylbird 263 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ (𝐵 = ∅ ∧ 𝐶 = ∅) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅))
5352imp 410 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)) = ∅)
5424, 53eqtr4d 2836 . . . . . 6 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ¬ (𝐵 = ∅ ∧ 𝐶 = ∅)) → (∅ ↑o (𝐵 +o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
5516, 54pm2.61dan 812 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ↑o (𝐵 +o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
56 oveq1 7143 . . . . . 6 (𝐴 = ∅ → (𝐴o (𝐵 +o 𝐶)) = (∅ ↑o (𝐵 +o 𝐶)))
57 oveq1 7143 . . . . . . 7 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
58 oveq1 7143 . . . . . . 7 (𝐴 = ∅ → (𝐴o 𝐶) = (∅ ↑o 𝐶))
5957, 58oveq12d 7154 . . . . . 6 (𝐴 = ∅ → ((𝐴o 𝐵) ·o (𝐴o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶)))
6056, 59eqeq12d 2814 . . . . 5 (𝐴 = ∅ → ((𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)) ↔ (∅ ↑o (𝐵 +o 𝐶)) = ((∅ ↑o 𝐵) ·o (∅ ↑o 𝐶))))
6155, 60syl5ibr 249 . . . 4 (𝐴 = ∅ → ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶))))
6261impcom 411 . . 3 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 = ∅) → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
63 oveq1 7143 . . . . . . . 8 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴o (𝐵 +o 𝐶)) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o (𝐵 +o 𝐶)))
64 oveq1 7143 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴o 𝐵) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐵))
65 oveq1 7143 . . . . . . . . 9 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴o 𝐶) = (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ↑o 𝐶))
6664, 65oveq12d 7154 . . . . . . . 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 2814 . . . . . . 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 344 . . . . . 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 7143 . . . . . . . . 9 (𝐵 = if(𝐵 ∈ On, 𝐵, 1o) → (𝐵 +o 𝐶) = (if(𝐵 ∈ On, 𝐵, 1o) +o 𝐶))
7069oveq2d 7152 . . . . . . . 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 7144 . . . . . . . . 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 7151 . . . . . . . 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 2814 . . . . . . 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 344 . . . . . 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 2877 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (𝐴 ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
76 eleq2 2878 . . . . . . . . . 10 (𝐴 = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 𝐴 ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
7775, 76anbi12d 633 . . . . . . . . 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 2877 . . . . . . . . . 10 (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (1o ∈ On ↔ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On))
79 eleq2 2878 . . . . . . . . . 10 (1o = if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) → (∅ ∈ 1o ↔ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)))
8078, 79anbi12d 633 . . . . . . . . 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 8095 . . . . . . . . . 10 1o ∈ On
82 0lt1o 8115 . . . . . . . . . 10 ∅ ∈ 1o
8381, 82pm3.2i 474 . . . . . . . . 9 (1o ∈ On ∧ ∅ ∈ 1o)
8477, 80, 83elimhyp 4488 . . . . . . . 8 (if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On ∧ ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o))
8584simpli 487 . . . . . . 7 if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o) ∈ On
8684simpri 489 . . . . . . 7 ∅ ∈ if((𝐴 ∈ On ∧ ∅ ∈ 𝐴), 𝐴, 1o)
8781elimel 4492 . . . . . . 7 if(𝐵 ∈ On, 𝐵, 1o) ∈ On
8885, 86, 87oeoalem 8208 . . . . . 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 4482 . . . . 5 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶))))
9089impr 458 . . . 4 (((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
9190an32s 651 . . 3 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
9262, 91oe0lem 8124 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝐶 ∈ On)) → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
93923impb 1112 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243  ifcif 4425  Oncon0 6160  (class class class)co 7136  1oc1o 8081   +o coa 8085   ·o comu 8086   ↑o coe 8087 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-omul 8093  df-oexp 8094 This theorem is referenced by:  oeoelem  8210  infxpenc  9432
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