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Theorem odi 8192
Description: Distributive law for ordinal arithmetic (left-distributivity). Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.)
Assertion
Ref Expression
odi ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Proof of Theorem odi
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7147 . . . . . 6 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
21oveq2d 7155 . . . . 5 (𝑥 = ∅ → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o ∅)))
3 oveq2 7147 . . . . . 6 (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
43oveq2d 7155 . . . . 5 (𝑥 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
52, 4eqeq12d 2817 . . . 4 (𝑥 = ∅ → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅))))
6 oveq2 7147 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7155 . . . . 5 (𝑥 = 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝑦)))
8 oveq2 7147 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
98oveq2d 7155 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))
107, 9eqeq12d 2817 . . . 4 (𝑥 = 𝑦 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))))
11 oveq2 7147 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1211oveq2d 7155 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o suc 𝑦)))
13 oveq2 7147 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
1413oveq2d 7155 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))
1512, 14eqeq12d 2817 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))
16 oveq2 7147 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1716oveq2d 7155 . . . . 5 (𝑥 = 𝐶 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝐶)))
18 oveq2 7147 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
1918oveq2d 7155 . . . . 5 (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
2017, 19eqeq12d 2817 . . . 4 (𝑥 = 𝐶 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
21 omcl 8148 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
22 oa0 8128 . . . . . 6 ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
2321, 22syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
24 om0 8129 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2524adantr 484 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o ∅) = ∅)
2625oveq2d 7155 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
27 oa0 8128 . . . . . . 7 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
2827adantl 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
2928oveq2d 7155 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 +o ∅)) = (𝐴 ·o 𝐵))
3023, 26, 293eqtr4rd 2847 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
31 oveq1 7146 . . . . . . . 8 ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴))
32 oasuc 8136 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
33323adant1 1127 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3433oveq2d 7155 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 +o suc 𝑦)) = (𝐴 ·o suc (𝐵 +o 𝑦)))
35 oacl 8147 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
36 omsuc 8138 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
3735, 36sylan2 595 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
38373impb 1112 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
3934, 38eqtrd 2836 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
40 omsuc 8138 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
41403adant2 1128 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
4241oveq2d 7155 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
43 omcl 8148 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑦) ∈ On)
44 oaass 8174 . . . . . . . . . . . . . . . . . 18 (((𝐴 ·o 𝐵) ∈ On ∧ (𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
4521, 44syl3an1 1160 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
4643, 45syl3an2 1161 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
47463exp 1116 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))
4847exp4b 434 . . . . . . . . . . . . . 14 (𝐴 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (𝑦 ∈ On → (𝐴 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))))
4948pm2.43a 54 . . . . . . . . . . . . 13 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))))
5049com4r 94 . . . . . . . . . . . 12 (𝐴 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))))
5150pm2.43i 52 . . . . . . . . . . 11 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))
52513imp 1108 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
5342, 52eqtr4d 2839 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴))
5439, 53eqeq12d 2817 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) ↔ ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴)))
5531, 54syl5ibr 249 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))
56553exp 1116 . . . . . 6 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))))
5756com3r 87 . . . . 5 (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))))
5857impd 414 . . . 4 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))))
59 vex 3447 . . . . . . . . . . . . . 14 𝑥 ∈ V
60 limelon 6226 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
6159, 60mpan 689 . . . . . . . . . . . . 13 (Lim 𝑥𝑥 ∈ On)
62 oacl 8147 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
63 om0r 8151 . . . . . . . . . . . . . . 15 ((𝐵 +o 𝑥) ∈ On → (∅ ·o (𝐵 +o 𝑥)) = ∅)
6462, 63syl 17 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∅ ·o (𝐵 +o 𝑥)) = ∅)
65 om0r 8151 . . . . . . . . . . . . . . . 16 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
66 om0r 8151 . . . . . . . . . . . . . . . 16 (𝑥 ∈ On → (∅ ·o 𝑥) = ∅)
6765, 66oveqan12d 7158 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∅ ·o 𝐵) +o (∅ ·o 𝑥)) = (∅ +o ∅))
68 0elon 6216 . . . . . . . . . . . . . . . 16 ∅ ∈ On
69 oa0 8128 . . . . . . . . . . . . . . . 16 (∅ ∈ On → (∅ +o ∅) = ∅)
7068, 69ax-mp 5 . . . . . . . . . . . . . . 15 (∅ +o ∅) = ∅
7167, 70eqtr2di 2853 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ∅ = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
7264, 71eqtrd 2836 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
7361, 72sylan2 595 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Lim 𝑥) → (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
7473ancoms 462 . . . . . . . . . . 11 ((Lim 𝑥𝐵 ∈ On) → (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
75 oveq1 7146 . . . . . . . . . . . 12 (𝐴 = ∅ → (𝐴 ·o (𝐵 +o 𝑥)) = (∅ ·o (𝐵 +o 𝑥)))
76 oveq1 7146 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
77 oveq1 7146 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐴 ·o 𝑥) = (∅ ·o 𝑥))
7876, 77oveq12d 7157 . . . . . . . . . . . 12 (𝐴 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥)))
7975, 78eqeq12d 2817 . . . . . . . . . . 11 (𝐴 = ∅ → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (∅ ·o (𝐵 +o 𝑥)) = ((∅ ·o 𝐵) +o (∅ ·o 𝑥))))
8074, 79syl5ibr 249 . . . . . . . . . 10 (𝐴 = ∅ → ((Lim 𝑥𝐵 ∈ On) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))
8180expd 419 . . . . . . . . 9 (𝐴 = ∅ → (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
8281com3r 87 . . . . . . . 8 (𝐵 ∈ On → (𝐴 = ∅ → (Lim 𝑥 → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
8382imp 410 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))
8483a1dd 50 . . . . . 6 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
85 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝐵 ∈ On)
8662ancoms 462 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
87 onelon 6188 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 +o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
8886, 87sylan 583 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
89 ontri1 6197 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ¬ 𝑧𝐵))
90 oawordex 8170 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧))
9189, 90bitr3d 284 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧𝐵 ↔ ∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧))
9285, 88, 91syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (¬ 𝑧𝐵 ↔ ∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧))
93 oaord 8160 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑣 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣𝑥 ↔ (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
94933expb 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣𝑥 ↔ (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
95 eleq1 2880 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 +o 𝑣) = 𝑧 → ((𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥) ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
9694, 95sylan9bb 513 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑣) = 𝑧) → (𝑣𝑥𝑧 ∈ (𝐵 +o 𝑥)))
97 iba 531 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 +o 𝑣) = 𝑧 → (𝑣𝑥 ↔ (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
9897adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑣) = 𝑧) → (𝑣𝑥 ↔ (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
9996, 98bitr3d 284 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑣) = 𝑧) → (𝑧 ∈ (𝐵 +o 𝑥) ↔ (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
10099an32s 651 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣 ∈ On ∧ (𝐵 +o 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +o 𝑥) ↔ (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
101100biimpcd 252 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ (𝐵 +o 𝑥) → (((𝑣 ∈ On ∧ (𝐵 +o 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
102101exp4c 436 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝐵 +o 𝑥) → (𝑣 ∈ On → ((𝐵 +o 𝑣) = 𝑧 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))))
103102com4r 94 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑣 ∈ On → ((𝐵 +o 𝑣) = 𝑧 → (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))))
104103imp 410 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑣 ∈ On → ((𝐵 +o 𝑣) = 𝑧 → (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧))))
105104reximdvai 3234 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (∃𝑣 ∈ On (𝐵 +o 𝑣) = 𝑧 → ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
10692, 105sylbid 243 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (¬ 𝑧𝐵 → ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
107106orrd 860 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
10861, 107sylanl1 679 . . . . . . . . . . . . . . . 16 (((Lim 𝑥𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
109108adantlrl 719 . . . . . . . . . . . . . . 15 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
110109adantlr 714 . . . . . . . . . . . . . 14 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)))
111 0ellim 6225 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Lim 𝑥 → ∅ ∈ 𝑥)
112 om00el 8189 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·o 𝑥) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥)))
113112biimprd 251 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥) → ∅ ∈ (𝐴 ·o 𝑥)))
114111, 113sylan2i 608 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((∅ ∈ 𝐴 ∧ Lim 𝑥) → ∅ ∈ (𝐴 ·o 𝑥)))
11561, 114sylan2 595 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ Lim 𝑥) → ((∅ ∈ 𝐴 ∧ Lim 𝑥) → ∅ ∈ (𝐴 ·o 𝑥)))
116115exp4b 434 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐴 → (Lim 𝑥 → ∅ ∈ (𝐴 ·o 𝑥)))))
117116com4r 94 . . . . . . . . . . . . . . . . . . . . . . 23 (Lim 𝑥 → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ·o 𝑥)))))
118117pm2.43a 54 . . . . . . . . . . . . . . . . . . . . . 22 (Lim 𝑥 → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ·o 𝑥))))
119118imp31 421 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ·o 𝑥))
120119a1d 25 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∅ ∈ (𝐴 ·o 𝑥)))
121120adantlrr 720 . . . . . . . . . . . . . . . . . . 19 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∅ ∈ (𝐴 ·o 𝑥)))
122 omordi 8179 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵)))
123122ancom1s 652 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵)))
124 onelss 6205 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o 𝐵)))
12522sseq2d 3950 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅) ↔ (𝐴 ·o 𝑧) ⊆ (𝐴 ·o 𝐵)))
126124, 125sylibrd 262 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
12721, 126syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
128127adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑧) ∈ (𝐴 ·o 𝐵) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
129123, 128syld 47 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
130129adantll 713 . . . . . . . . . . . . . . . . . . 19 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
131121, 130jcad 516 . . . . . . . . . . . . . . . . . 18 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (∅ ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅))))
132 oveq2 7147 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ∅ → ((𝐴 ·o 𝐵) +o 𝑤) = ((𝐴 ·o 𝐵) +o ∅))
133132sseq2d 3950 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ∅ → ((𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤) ↔ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)))
134133rspcev 3574 . . . . . . . . . . . . . . . . . 18 ((∅ ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o ∅)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
135131, 134syl6 35 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
136135adantrr 716 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (𝑧𝐵 → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
137 omordi 8179 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑣𝑥 → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
13861, 137sylanl1 679 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑣𝑥 → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
139138adantrd 495 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
140139adantrr 716 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥)))
141 oveq2 7147 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑣 → (𝐵 +o 𝑦) = (𝐵 +o 𝑣))
142141oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → (𝐴 ·o (𝐵 +o 𝑦)) = (𝐴 ·o (𝐵 +o 𝑣)))
143 oveq2 7147 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑣 → (𝐴 ·o 𝑦) = (𝐴 ·o 𝑣))
144143oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
145142, 144eqeq12d 2817 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑣 → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) ↔ (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
146145rspccv 3571 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝑣𝑥 → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
147 oveq2 7147 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 +o 𝑣) = 𝑧 → (𝐴 ·o (𝐵 +o 𝑣)) = (𝐴 ·o 𝑧))
148 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) → ((𝐴 ·o (𝐵 +o 𝑣)) = (𝐴 ·o 𝑧) ↔ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) = (𝐴 ·o 𝑧)))
149147, 148syl5ib 247 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) → ((𝐵 +o 𝑣) = 𝑧 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) = (𝐴 ·o 𝑧)))
150 eqimss2 3975 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) = (𝐴 ·o 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
151149, 150syl6 35 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)) → ((𝐵 +o 𝑣) = 𝑧 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
152151imim2i 16 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣𝑥 → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))) → (𝑣𝑥 → ((𝐵 +o 𝑣) = 𝑧 → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))))
153152impd 414 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣𝑥 → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))) → ((𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
154146, 153syl 17 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → ((𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
155154ad2antll 728 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
156140, 155jcad 516 . . . . . . . . . . . . . . . . . . 19 (((Lim 𝑥𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ((𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))))
157 oveq2 7147 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
158157sseq2d 3950 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐴 ·o 𝑣) → ((𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤) ↔ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
159158rspcev 3574 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ·o 𝑣) ∈ (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
160156, 159syl6 35 . . . . . . . . . . . . . . . . . 18 (((Lim 𝑥𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
161160rexlimdvw 3252 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥𝐴 ∈ On) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
162161adantlrr 720 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
163136, 162jaod 856 . . . . . . . . . . . . . . 15 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
164163adantr 484 . . . . . . . . . . . . . 14 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ((𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +o 𝑣) = 𝑧)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤)))
165110, 164mpd 15 . . . . . . . . . . . . 13 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
166165ralrimiva 3152 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤))
167 iunss2 4939 . . . . . . . . . . . 12 (∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑤 ∈ (𝐴 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ ((𝐴 ·o 𝐵) +o 𝑤) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧) ⊆ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
168166, 167syl 17 . . . . . . . . . . 11 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧) ⊆ 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
169 omordlim 8190 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
170169ex 416 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑤 ∈ (𝐴 ·o 𝑥) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣)))
17159, 170mpanr1 702 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝑤 ∈ (𝐴 ·o 𝑥) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣)))
172171ancoms 462 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥𝐴 ∈ On) → (𝑤 ∈ (𝐴 ·o 𝑥) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣)))
173172imp 410 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥𝐴 ∈ On) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
174173adantlrr 720 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
175174adantlr 714 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣))
176 oaordi 8159 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣𝑥 → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
17761, 176sylan 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((Lim 𝑥𝐵 ∈ On) → (𝑣𝑥 → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
178177imp 410 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥𝐵 ∈ On) ∧ 𝑣𝑥) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥))
179178adantlrl 719 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣𝑥) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥))
180179a1d 25 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
181180adantlr 714 . . . . . . . . . . . . . . . . . . 19 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → (𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥)))
182 limord 6222 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Lim 𝑥 → Ord 𝑥)
183 ordelon 6187 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Ord 𝑥𝑣𝑥) → 𝑣 ∈ On)
184182, 183sylan 583 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Lim 𝑥𝑣𝑥) → 𝑣 ∈ On)
185 omcl 8148 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ On ∧ 𝑣 ∈ On) → (𝐴 ·o 𝑣) ∈ On)
186185ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝑣) ∈ On)
187186adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o 𝑣) ∈ On)
18821adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o 𝐵) ∈ On)
189 oaordi 8159 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴 ·o 𝑣) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
190187, 188, 189syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
191184, 190sylan 583 . . . . . . . . . . . . . . . . . . . . . . 23 (((Lim 𝑥𝑣𝑥) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
192191an32s 651 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
193192adantlr 714 . . . . . . . . . . . . . . . . . . . . 21 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
194145rspccva 3573 . . . . . . . . . . . . . . . . . . . . . . 23 ((∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) ∧ 𝑣𝑥) → (𝐴 ·o (𝐵 +o 𝑣)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣)))
195194eleq2d 2878 . . . . . . . . . . . . . . . . . . . . . 22 ((∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) ∧ 𝑣𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) ↔ ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
196195adantll 713 . . . . . . . . . . . . . . . . . . . . 21 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) ↔ ((𝐴 ·o 𝐵) +o 𝑤) ∈ ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑣))))
197193, 196sylibrd 262 . . . . . . . . . . . . . . . . . . . 20 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣))))
198 oacl 8147 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑣 ∈ On) → (𝐵 +o 𝑣) ∈ On)
199198ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑣) ∈ On)
200 omcl 8148 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ∈ On ∧ (𝐵 +o 𝑣) ∈ On) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
201199, 200sylan2 595 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ (𝑣 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
202201an12s 648 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
203184, 202sylan 583 . . . . . . . . . . . . . . . . . . . . . . 23 (((Lim 𝑥𝑣𝑥) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
204203an32s 651 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣𝑥) → (𝐴 ·o (𝐵 +o 𝑣)) ∈ On)
205 onelss 6205 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ·o (𝐵 +o 𝑣)) ∈ On → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
206204, 205syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
207206adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (((𝐴 ·o 𝐵) +o 𝑤) ∈ (𝐴 ·o (𝐵 +o 𝑣)) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
208197, 207syld 47 . . . . . . . . . . . . . . . . . . 19 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
209181, 208jcad 516 . . . . . . . . . . . . . . . . . 18 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ((𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥) ∧ ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣)))))
210 oveq2 7147 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 +o 𝑣) → (𝐴 ·o 𝑧) = (𝐴 ·o (𝐵 +o 𝑣)))
211210sseq2d 3950 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +o 𝑣) → (((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧) ↔ ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))))
212211rspcev 3574 . . . . . . . . . . . . . . . . . 18 (((𝐵 +o 𝑣) ∈ (𝐵 +o 𝑥) ∧ ((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o (𝐵 +o 𝑣))) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧))
213209, 212syl6 35 . . . . . . . . . . . . . . . . 17 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑣𝑥) → (𝑤 ∈ (𝐴 ·o 𝑣) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧)))
214213rexlimdva 3246 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) → (∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧)))
215214adantr 484 . . . . . . . . . . . . . . 15 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → (∃𝑣𝑥 𝑤 ∈ (𝐴 ·o 𝑣) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧)))
216175, 215mpd 15 . . . . . . . . . . . . . 14 ((((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) ∧ 𝑤 ∈ (𝐴 ·o 𝑥)) → ∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧))
217216ralrimiva 3152 . . . . . . . . . . . . 13 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) → ∀𝑤 ∈ (𝐴 ·o 𝑥)∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧))
218 iunss2 4939 . . . . . . . . . . . . 13 (∀𝑤 ∈ (𝐴 ·o 𝑥)∃𝑧 ∈ (𝐵 +o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ (𝐴 ·o 𝑧) → 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
219217, 218syl 17 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) → 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
220219adantrl 715 . . . . . . . . . . 11 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
221168, 220eqssd 3935 . . . . . . . . . 10 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧) = 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
222 oalimcl 8173 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +o 𝑥))
22359, 222mpanr1 702 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +o 𝑥))
224223ancoms 462 . . . . . . . . . . . . . 14 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +o 𝑥))
225224anim2i 619 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)))
226225an12s 648 . . . . . . . . . . . 12 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)))
227 ovex 7172 . . . . . . . . . . . . 13 (𝐵 +o 𝑥) ∈ V
228 omlim 8145 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ ((𝐵 +o 𝑥) ∈ V ∧ Lim (𝐵 +o 𝑥))) → (𝐴 ·o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
229227, 228mpanr1 702 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)) → (𝐴 ·o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
230226, 229syl 17 . . . . . . . . . . 11 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
231230adantr 484 . . . . . . . . . 10 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (𝐴 ·o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 ·o 𝑧))
23221ad2antlr 726 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴 ·o 𝐵) ∈ On)
23359jctl 527 . . . . . . . . . . . . . . . 16 (Lim 𝑥 → (𝑥 ∈ V ∧ Lim 𝑥))
234233anim1ci 618 . . . . . . . . . . . . . . 15 ((Lim 𝑥𝐴 ∈ On) → (𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)))
235 omlimcl 8191 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝑥))
236234, 235sylan 583 . . . . . . . . . . . . . 14 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝑥))
237236adantlrr 720 . . . . . . . . . . . . 13 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝑥))
238 ovex 7172 . . . . . . . . . . . . 13 (𝐴 ·o 𝑥) ∈ V
239237, 238jctil 523 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) ∈ V ∧ Lim (𝐴 ·o 𝑥)))
240 oalim 8144 . . . . . . . . . . . 12 (((𝐴 ·o 𝐵) ∈ On ∧ ((𝐴 ·o 𝑥) ∈ V ∧ Lim (𝐴 ·o 𝑥))) → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
241232, 239, 240syl2anc 587 . . . . . . . . . . 11 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
242241adantrr 716 . . . . . . . . . 10 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = 𝑤 ∈ (𝐴 ·o 𝑥)((𝐴 ·o 𝐵) +o 𝑤))
243221, 231, 2423eqtr4d 2846 . . . . . . . . 9 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ (∅ ∈ 𝐴 ∧ ∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))
244243exp43 440 . . . . . . . 8 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))))
245244com3l 89 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (Lim 𝑥 → (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))))
246245imp 410 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
24784, 246oe0lem 8125 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Lim 𝑥 → (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
248247com12 32 . . . 4 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))))
2495, 10, 15, 20, 30, 58, 248tfinds3 7563 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
250249expdcom 418 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))))
2512503imp 1108 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  wss 3884  c0 4246   ciun 4884  Ord word 6162  Oncon0 6163  Lim wlim 6164  suc csuc 6165  (class class class)co 7139   +o coa 8086   ·o comu 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094
This theorem is referenced by:  omass  8193  oeeui  8215  oaabs2  8259
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