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Theorem odi 7546
 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 6557 . . . . . 6 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
21oveq2d 6565 . . . . 5 (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 ∅)))
3 oveq2 6557 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
43oveq2d 6565 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
52, 4eqeq12d 2625 . . . 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 6557 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6565 . . . . 5 (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)))
8 oveq2 6557 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
98oveq2d 6565 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)))
107, 9eqeq12d 2625 . . . 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 6557 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1211oveq2d 6565 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)))
13 oveq2 6557 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
1413oveq2d 6565 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)))
1512, 14eqeq12d 2625 . . . 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 6557 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1716oveq2d 6565 . . . . 5 (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
18 oveq2 6557 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
1918oveq2d 6565 . . . . 5 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
2017, 19eqeq12d 2625 . . . 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 7503 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
22 oa0 7483 . . . . . 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 7484 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
2524adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 ∅) = ∅)
2625oveq2d 6565 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
27 oa0 7483 . . . . . . 7 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
2827adantl 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
2928oveq2d 6565 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = (𝐴 ·𝑜 𝐵))
3023, 26, 293eqtr4rd 2655 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
31 oveq1 6556 . . . . . . . 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 7491 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
33323adant1 1072 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3433oveq2d 6565 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)))
35 oacl 7502 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
36 omsuc 7493 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
3735, 36sylan2 490 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
38373impb 1252 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
3934, 38eqtrd 2644 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
40 omsuc 7493 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
41403adant2 1073 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4241oveq2d 6565 . . . . . . . . . 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 7503 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
44 oaass 7528 . . . . . . . . . . . . . . . . . 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 1351 . . . . . . . . . . . . . . . . 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 1352 . . . . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . 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 52 . . . . . . . . . . . . 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 92 . . . . . . . . . . . 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 50 . . . . . . . . . . 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 1249 . . . . . . . . . 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 2647 . . . . . . . . 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 2625 . . . . . . . 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 235 . . . . . . 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 1256 . . . . . 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 85 . . . . 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 446 . . . 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 3176 . . . . . . . . . . . . . 14 𝑥 ∈ V
60 limelon 5705 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
6159, 60mpan 702 . . . . . . . . . . . . 13 (Lim 𝑥𝑥 ∈ On)
62 oacl 7502 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
63 om0r 7506 . . . . . . . . . . . . . . 15 ((𝐵 +𝑜 𝑥) ∈ On → (∅ ·𝑜 (𝐵 +𝑜 𝑥)) = ∅)
6462, 63syl 17 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∅ ·𝑜 (𝐵 +𝑜 𝑥)) = ∅)
65 om0r 7506 . . . . . . . . . . . . . . . 16 (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
66 om0r 7506 . . . . . . . . . . . . . . . 16 (𝑥 ∈ On → (∅ ·𝑜 𝑥) = ∅)
6765, 66oveqan12d 6568 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)) = (∅ +𝑜 ∅))
68 0elon 5695 . . . . . . . . . . . . . . . 16 ∅ ∈ On
69 oa0 7483 . . . . . . . . . . . . . . . 16 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
7068, 69ax-mp 5 . . . . . . . . . . . . . . 15 (∅ +𝑜 ∅) = ∅
7167, 70syl6req 2661 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ∅ = ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)))
7264, 71eqtrd 2644 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∅ ·𝑜 (𝐵 +𝑜 𝑥)) = ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)))
7361, 72sylan2 490 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Lim 𝑥) → (∅ ·𝑜 (𝐵 +𝑜 𝑥)) = ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)))
7473ancoms 468 . . . . . . . . . . 11 ((Lim 𝑥𝐵 ∈ On) → (∅ ·𝑜 (𝐵 +𝑜 𝑥)) = ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)))
75 oveq1 6556 . . . . . . . . . . . 12 (𝐴 = ∅ → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (∅ ·𝑜 (𝐵 +𝑜 𝑥)))
76 oveq1 6556 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
77 oveq1 6556 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐴 ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
7876, 77oveq12d 6567 . . . . . . . . . . . 12 (𝐴 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)))
7975, 78eqeq12d 2625 . . . . . . . . . . 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 235 . . . . . . . . . 10 (𝐴 = ∅ → ((Lim 𝑥𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥))))
8180expd 451 . . . . . . . . 9 (𝐴 = ∅ → (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)))))
8281com3r 85 . . . . . . . 8 (𝐵 ∈ On → (𝐴 = ∅ → (Lim 𝑥 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)))))
8382imp 444 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (Lim 𝑥 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥))))
8483a1dd 48 . . . . . 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 788 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝐵 ∈ On)
8662ancoms 468 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
87 onelon 5665 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 +𝑜 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On)
8886, 87sylan 487 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On)
89 ontri1 5674 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ¬ 𝑧𝐵))
90 oawordex 7524 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧))
9189, 90bitr3d 269 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧𝐵 ↔ ∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧))
9285, 88, 91syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (¬ 𝑧𝐵 ↔ ∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧))
93 oaord 7514 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑣 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣𝑥 ↔ (𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥)))
94933expb 1258 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣𝑥 ↔ (𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥)))
95 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 +𝑜 𝑣) = 𝑧 → ((𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥) ↔ 𝑧 ∈ (𝐵 +𝑜 𝑥)))
9694, 95sylan9bb 732 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +𝑜 𝑣) = 𝑧) → (𝑣𝑥𝑧 ∈ (𝐵 +𝑜 𝑥)))
97 iba 523 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 +𝑜 𝑣) = 𝑧 → (𝑣𝑥 ↔ (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
9897adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +𝑜 𝑣) = 𝑧) → (𝑣𝑥 ↔ (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
9996, 98bitr3d 269 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +𝑜 𝑣) = 𝑧) → (𝑧 ∈ (𝐵 +𝑜 𝑥) ↔ (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
10099an32s 842 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣 ∈ On ∧ (𝐵 +𝑜 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +𝑜 𝑥) ↔ (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
101100biimpcd 238 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ (𝐵 +𝑜 𝑥) → (((𝑣 ∈ On ∧ (𝐵 +𝑜 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
102101exp4c 634 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝐵 +𝑜 𝑥) → (𝑣 ∈ On → ((𝐵 +𝑜 𝑣) = 𝑧 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))))
103102com4r 92 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → (𝑣 ∈ On → ((𝐵 +𝑜 𝑣) = 𝑧 → (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))))
104103imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑣 ∈ On → ((𝐵 +𝑜 𝑣) = 𝑧 → (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧))))
105104reximdvai 2998 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧 → ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
10692, 105sylbid 229 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (¬ 𝑧𝐵 → ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
107106orrd 392 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
10861, 107sylanl1 680 . . . . . . . . . . . . . . . 16 (((Lim 𝑥𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
109108adantlrl 752 . . . . . . . . . . . . . . 15 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
110109adantlr 747 . . . . . . . . . . . . . 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 5704 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Lim 𝑥 → ∅ ∈ 𝑥)
112 om00el 7543 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝑥) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥)))
113112biimprd 237 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥) → ∅ ∈ (𝐴 ·𝑜 𝑥)))
114111, 113sylan2i 685 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((∅ ∈ 𝐴 ∧ Lim 𝑥) → ∅ ∈ (𝐴 ·𝑜 𝑥)))
11561, 114sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ Lim 𝑥) → ((∅ ∈ 𝐴 ∧ Lim 𝑥) → ∅ ∈ (𝐴 ·𝑜 𝑥)))
116115exp4b 630 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐴 → (Lim 𝑥 → ∅ ∈ (𝐴 ·𝑜 𝑥)))))
117116com4r 92 . . . . . . . . . . . . . . . . . . . . . . 23 (Lim 𝑥 → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ·𝑜 𝑥)))))
118117pm2.43a 52 . . . . . . . . . . . . . . . . . . . . . 22 (Lim 𝑥 → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ·𝑜 𝑥))))
119118imp31 447 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ·𝑜 𝑥))
120119a1d 25 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∅ ∈ (𝐴 ·𝑜 𝑥)))
121120adantlrr 753 . . . . . . . . . . . . . . . . . . 19 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∅ ∈ (𝐴 ·𝑜 𝑥)))
122 omordi 7533 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 𝐵)))
123122ancom1s 843 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 𝐵)))
124 onelss 5683 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ·𝑜 𝐵) ∈ On → ((𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 𝐵) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 𝐵)))
12522sseq2d 3596 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ·𝑜 𝐵) ∈ On → ((𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅) ↔ (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 𝐵)))
126124, 125sylibrd 248 . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 𝐵) → (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅)))
129123, 128syld 46 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅)))
130129adantll 746 . . . . . . . . . . . . . . . . . . 19 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅)))
131121, 130jcad 554 . . . . . . . . . . . . . . . . . 18 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (∅ ∈ (𝐴 ·𝑜 𝑥) ∧ (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅))))
132 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
133132sseq2d 3596 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ∅ → ((𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ↔ (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅)))
134133rspcev 3282 . . . . . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∃𝑤 ∈ (𝐴 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝑤)))
136135adantrr 749 . . . . . . . . . . . . . . . 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 7533 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑣𝑥 → (𝐴 ·𝑜 𝑣) ∈ (𝐴 ·𝑜 𝑥)))
13861, 137sylanl1 680 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑣𝑥 → (𝐴 ·𝑜 𝑣) ∈ (𝐴 ·𝑜 𝑥)))
139138adantrd 483 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧) → (𝐴 ·𝑜 𝑣) ∈ (𝐴 ·𝑜 𝑥)))
140139adantrr 749 . . . . . . . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑣 → (𝐵 +𝑜 𝑦) = (𝐵 +𝑜 𝑣))
142141oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)))
143 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑣 → (𝐴 ·𝑜 𝑦) = (𝐴 ·𝑜 𝑣))
144143oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣)))
145142, 144eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . 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 3279 . . . . . . . . . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 +𝑜 𝑣) = 𝑧 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) = (𝐴 ·𝑜 𝑧))
148 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣)) = (𝐴 ·𝑜 𝑧) → (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣)))
151149, 150syl6 34 . . . . . . . . . . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐴 ·𝑜 𝑣) → ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣)))
158157sseq2d 3596 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐴 ·𝑜 𝑣) → ((𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ↔ (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣))))
159158rspcev 3282 . . . . . . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . . . . . . 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 3016 . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . 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 394 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 2949 . . . . . . . . . . . 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 4501 . . . . . . . . . . . 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 7544 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑤 ∈ (𝐴 ·𝑜 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·𝑜 𝑣))
170169ex 449 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑤 ∈ (𝐴 ·𝑜 𝑥) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·𝑜 𝑣)))
17159, 170mpanr1 715 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝑤 ∈ (𝐴 ·𝑜 𝑥) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·𝑜 𝑣)))
172171ancoms 468 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥𝐴 ∈ On) → (𝑤 ∈ (𝐴 ·𝑜 𝑥) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·𝑜 𝑣)))
173172imp 444 . . . . . . . . . . . . . . . . 17 (((Lim 𝑥𝐴 ∈ On) ∧ 𝑤 ∈ (𝐴 ·𝑜 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·𝑜 𝑣))
174173adantlrr 753 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑤 ∈ (𝐴 ·𝑜 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·𝑜 𝑣))
175174adantlr 747 . . . . . . . . . . . . . . 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 7513 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣𝑥 → (𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥)))
17761, 176sylan 487 . . . . . . . . . . . . . . . . . . . . . . 23 ((Lim 𝑥𝐵 ∈ On) → (𝑣𝑥 → (𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥)))
178177imp 444 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥𝐵 ∈ On) ∧ 𝑣𝑥) → (𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥))
179178adantlrl 752 . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Lim 𝑥 → Ord 𝑥)
183 ordelon 5664 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Ord 𝑥𝑣𝑥) → 𝑣 ∈ On)
184182, 183sylan 487 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Lim 𝑥𝑣𝑥) → 𝑣 ∈ On)
185 omcl 7503 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ On ∧ 𝑣 ∈ On) → (𝐴 ·𝑜 𝑣) ∈ On)
186185ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝑣) ∈ On)
187186adantrr 749 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 𝑣) ∈ On)
18821adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 𝐵) ∈ On)
189 oaordi 7513 . . . . . . . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . . . . 23 (((Lim 𝑥𝑣𝑥) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝑤 ∈ (𝐴 ·𝑜 𝑣) → ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ∈ ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣))))
192191an32s 842 . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . . 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 3281 . . . . . . . . . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . . . . . 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 7502 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑣 ∈ On) → (𝐵 +𝑜 𝑣) ∈ On)
199198ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝑣) ∈ On)
200 omcl 7503 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑣) ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) ∈ On)
201199, 200sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ (𝑣 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) ∈ On)
202201an12s 839 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑣 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) ∈ On)
203184, 202sylan 487 . . . . . . . . . . . . . . . . . . . . . . 23 (((Lim 𝑥𝑣𝑥) ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) ∈ On)
204203an32s 842 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣𝑥) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) ∈ On)
205 onelss 5683 . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . 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 46 . . . . . . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 +𝑜 𝑣) → (𝐴 ·𝑜 𝑧) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)))
211210sseq2d 3596 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +𝑜 𝑣) → (((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ⊆ (𝐴 ·𝑜 𝑧) ↔ ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ⊆ (𝐴 ·𝑜 (𝐵 +𝑜 𝑣))))
212211rspcev 3282 . . . . . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . . . . . 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 3013 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 2949 . . . . . . . . . . . . 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 4501 . . . . . . . . . . . . 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 748 . . . . . . . . . . 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 3585 . . . . . . . . . 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 7527 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +𝑜 𝑥))
22359, 222mpanr1 715 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +𝑜 𝑥))
224223ancoms 468 . . . . . . . . . . . . . 14 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +𝑜 𝑥))
225224anim2i 591 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +𝑜 𝑥)))
226225an12s 839 . . . . . . . . . . . 12 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +𝑜 𝑥)))
227 ovex 6577 . . . . . . . . . . . . 13 (𝐵 +𝑜 𝑥) ∈ V
228 omlim 7500 . . . . . . . . . . . . 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 715 . . . . . . . . . . . 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 480 . . . . . . . . . 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 759 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 𝐵) ∈ On)
23359jctl 562 . . . . . . . . . . . . . . . . 17 (Lim 𝑥 → (𝑥 ∈ V ∧ Lim 𝑥))
234233anim2i 591 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)))
235234ancoms 468 . . . . . . . . . . . . . . 15 ((Lim 𝑥𝐴 ∈ On) → (𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)))
236 omlimcl 7545 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝑥))
237235, 236sylan 487 . . . . . . . . . . . . . 14 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝑥))
238237adantlrr 753 . . . . . . . . . . . . 13 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝑥))
239 ovex 6577 . . . . . . . . . . . . 13 (𝐴 ·𝑜 𝑥) ∈ V
240238, 239jctil 558 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) ∈ V ∧ Lim (𝐴 ·𝑜 𝑥)))
241 oalim 7499 . . . . . . . . . . . 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))
242232, 240, 241syl2anc 691 . . . . . . . . . . 11 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = 𝑤 ∈ (𝐴 ·𝑜 𝑥)((𝐴 ·𝑜 𝐵) +𝑜 𝑤))
243242adantrr 749 . . . . . . . . . 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))
244221, 231, 2433eqtr4d 2654 . . . . . . . . 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)))
245244exp43 638 . . . . . . . 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))))))
246245com3l 87 . . . . . . 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))))))
247246imp 444 . . . . . 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)))))
24884, 247oe0lem 7480 . . . . 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)))))
249248com12 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)))))
2505, 10, 15, 20, 30, 58, 249tfinds3 6956 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶))))
251250expdcom 454 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))))
2522513imp 1249 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +𝑜 coa 7444   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452 This theorem is referenced by:  omass  7547  oeeui  7569  oaabs2  7612
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