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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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