MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odi Structured version   Visualization version   GIF version

Theorem odi 7704
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 6698 . . . . . 6 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
21oveq2d 6706 . . . . 5 (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 ∅)))
3 oveq2 6698 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
43oveq2d 6706 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
52, 4eqeq12d 2666 . . . 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 6698 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6706 . . . . 5 (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)))
8 oveq2 6698 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
98oveq2d 6706 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)))
107, 9eqeq12d 2666 . . . 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 6698 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1211oveq2d 6706 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)))
13 oveq2 6698 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
1413oveq2d 6706 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)))
1512, 14eqeq12d 2666 . . . 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 6698 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1716oveq2d 6706 . . . . 5 (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
18 oveq2 6698 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
1918oveq2d 6706 . . . . 5 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
2017, 19eqeq12d 2666 . . . 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 7661 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
22 oa0 7641 . . . . . 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 7642 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
2524adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 ∅) = ∅)
2625oveq2d 6706 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
27 oa0 7641 . . . . . . 7 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
2827adantl 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
2928oveq2d 6706 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = (𝐴 ·𝑜 𝐵))
3023, 26, 293eqtr4rd 2696 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
31 oveq1 6697 . . . . . . . 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 7649 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
33323adant1 1099 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3433oveq2d 6706 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)))
35 oacl 7660 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
36 omsuc 7651 . . . . . . . . . . . 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 1279 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
3934, 38eqtrd 2685 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
40 omsuc 7651 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
41403adant2 1100 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4241oveq2d 6706 . . . . . . . . . 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 7661 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
44 oaass 7686 . . . . . . . . . . . . . . . . . 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 1399 . . . . . . . . . . . . . . . . 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 1400 . . . . . . . . . . . . . . . 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 1283 . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . 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 1275 . . . . . . . . . 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 2688 . . . . . . . . 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 2666 . . . . . . . 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 236 . . . . . . 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 1283 . . . . . 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 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 3234 . . . . . . . . . . . . . 14 𝑥 ∈ V
60 limelon 5826 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
6159, 60mpan 706 . . . . . . . . . . . . 13 (Lim 𝑥𝑥 ∈ On)
62 oacl 7660 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
63 om0r 7664 . . . . . . . . . . . . . . 15 ((𝐵 +𝑜 𝑥) ∈ On → (∅ ·𝑜 (𝐵 +𝑜 𝑥)) = ∅)
6462, 63syl 17 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∅ ·𝑜 (𝐵 +𝑜 𝑥)) = ∅)
65 om0r 7664 . . . . . . . . . . . . . . . 16 (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
66 om0r 7664 . . . . . . . . . . . . . . . 16 (𝑥 ∈ On → (∅ ·𝑜 𝑥) = ∅)
6765, 66oveqan12d 6709 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)) = (∅ +𝑜 ∅))
68 0elon 5816 . . . . . . . . . . . . . . . 16 ∅ ∈ On
69 oa0 7641 . . . . . . . . . . . . . . . 16 (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
7068, 69ax-mp 5 . . . . . . . . . . . . . . 15 (∅ +𝑜 ∅) = ∅
7167, 70syl6req 2702 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ∅ = ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)))
7264, 71eqtrd 2685 . . . . . . . . . . . . 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 6697 . . . . . . . . . . . 12 (𝐴 = ∅ → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (∅ ·𝑜 (𝐵 +𝑜 𝑥)))
76 oveq1 6697 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
77 oveq1 6697 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐴 ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
7876, 77oveq12d 6708 . . . . . . . . . . . 12 (𝐴 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((∅ ·𝑜 𝐵) +𝑜 (∅ ·𝑜 𝑥)))
7975, 78eqeq12d 2666 . . . . . . . . . . 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 236 . . . . . . . . . 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 87 . . . . . . . 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 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 807 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝐵 ∈ On)
8662ancoms 468 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
87 onelon 5786 . . . . . . . . . . . . . . . . . . . . 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 5795 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ¬ 𝑧𝐵))
90 oawordex 7682 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧))
9189, 90bitr3d 270 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑧𝐵 ↔ ∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧))
9285, 88, 91syl2anc 694 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (¬ 𝑧𝐵 ↔ ∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧))
93 oaord 7672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑣 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑣𝑥 ↔ (𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥)))
94933expb 1285 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣𝑥 ↔ (𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥)))
95 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 +𝑜 𝑣) = 𝑧 → ((𝐵 +𝑜 𝑣) ∈ (𝐵 +𝑜 𝑥) ↔ 𝑧 ∈ (𝐵 +𝑜 𝑥)))
9694, 95sylan9bb 736 . . . . . . . . . . . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑣 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +𝑜 𝑣) = 𝑧) → (𝑧 ∈ (𝐵 +𝑜 𝑥) ↔ (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
10099an32s 863 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣 ∈ On ∧ (𝐵 +𝑜 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +𝑜 𝑥) ↔ (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
101100biimpcd 239 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ (𝐵 +𝑜 𝑥) → (((𝑣 ∈ On ∧ (𝐵 +𝑜 𝑣) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
102101exp4c 635 . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑣 ∈ On → ((𝐵 +𝑜 𝑣) = 𝑧 → (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧))))
105104reximdvai 3044 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (∃𝑣 ∈ On (𝐵 +𝑜 𝑣) = 𝑧 → ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
10692, 105sylbid 230 . . . . . . . . . . . . . . . . . 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 683 . . . . . . . . . . . . . . . 16 (((Lim 𝑥𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
109108adantlrl 756 . . . . . . . . . . . . . . 15 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧𝐵 ∨ ∃𝑣 ∈ On (𝑣𝑥 ∧ (𝐵 +𝑜 𝑣) = 𝑧)))
110109adantlr 751 . . . . . . . . . . . . . 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 5825 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Lim 𝑥 → ∅ ∈ 𝑥)
112 om00el 7701 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝑥) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥)))
113112biimprd 238 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥) → ∅ ∈ (𝐴 ·𝑜 𝑥)))
114111, 113sylan2i 688 . . . . . . . . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ·𝑜 𝑥))
120119a1d 25 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∅ ∈ (𝐴 ·𝑜 𝑥)))
121120adantlrr 757 . . . . . . . . . . . . . . . . . . 19 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → ∅ ∈ (𝐴 ·𝑜 𝑥)))
122 omordi 7691 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 𝐵)))
123122ancom1s 864 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 𝐵)))
124 onelss 5804 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ·𝑜 𝐵) ∈ On → ((𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 𝐵) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 𝐵)))
12522sseq2d 3666 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ·𝑜 𝐵) ∈ On → ((𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅) ↔ (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 𝐵)))
126124, 125sylibrd 249 . . . . . . . . . . . . . . . . . . . . . . 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 47 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧𝐵 → (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅)))
130129adantll 750 . . . . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
133132sseq2d 3666 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ∅ → ((𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ↔ (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 ∅)))
134133rspcev 3340 . . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . 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 7691 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑣𝑥 → (𝐴 ·𝑜 𝑣) ∈ (𝐴 ·𝑜 𝑥)))
13861, 137sylanl1 683 . . . . . . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑣 → (𝐵 +𝑜 𝑦) = (𝐵 +𝑜 𝑣))
142141oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)))
143 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑣 → (𝐴 ·𝑜 𝑦) = (𝐴 ·𝑜 𝑣))
144143oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣)))
145142, 144eqeq12d 2666 . . . . . . . . . . . . . . . . . . . . . . 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 3337 . . . . . . . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 +𝑜 𝑣) = 𝑧 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) = (𝐴 ·𝑜 𝑧))
148 eqeq1 2655 . . . . . . . . . . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . . . . . . . . . . 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 3691 . . . . . . . . . . . . . . . . . . . . . . . . 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 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 765 . . . . . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐴 ·𝑜 𝑣) → ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣)))
158157sseq2d 3666 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐴 ·𝑜 𝑣) → ((𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ↔ (𝐴 ·𝑜 𝑧) ⊆ ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑣))))
159158rspcev 3340 . . . . . . . . . . . . . . . . . . 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 3063 . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 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 2995 . . . . . . . . . . . 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 4597 . . . . . . . . . . . 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 7702 . . . . . . . . . . . . . . . . . . . . 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 719 . . . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 16 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑤 ∈ (𝐴 ·𝑜 𝑥)) → ∃𝑣𝑥 𝑤 ∈ (𝐴 ·𝑜 𝑣))
175174adantlr 751 . . . . . . . . . . . . . . 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 7671 . . . . . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . . . . . . 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 5822 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Lim 𝑥 → Ord 𝑥)
183 ordelon 5785 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Ord 𝑥𝑣𝑥) → 𝑣 ∈ On)
184182, 183sylan 487 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Lim 𝑥𝑣𝑥) → 𝑣 ∈ On)
185 omcl 7661 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ On ∧ 𝑣 ∈ On) → (𝐴 ·𝑜 𝑣) ∈ On)
186185ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝑣) ∈ On)
187186adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . . 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 7671 . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . 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 863 . . . . . . . . . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . . . . . . . . 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 3339 . . . . . . . . . . . . . . . . . . . . . . 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 2716 . . . . . . . . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . . . . . 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 7660 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑣 ∈ On) → (𝐵 +𝑜 𝑣) ∈ On)
199198ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝑣) ∈ On)
200 omcl 7661 . . . . . . . . . . . . . . . . . . . . . . . . . 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 860 . . . . . . . . . . . . . . . . . . . . . . . 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 863 . . . . . . . . . . . . . . . . . . . . . 22 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑣𝑥) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)) ∈ On)
205 onelss 5804 . . . . . . . . . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . . . . . . . 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 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 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 +𝑜 𝑣) → (𝐴 ·𝑜 𝑧) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑣)))
211210sseq2d 3666 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +𝑜 𝑣) → (((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ⊆ (𝐴 ·𝑜 𝑧) ↔ ((𝐴 ·𝑜 𝐵) +𝑜 𝑤) ⊆ (𝐴 ·𝑜 (𝐵 +𝑜 𝑣))))
212211rspcev 3340 . . . . . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . . . . 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 2995 . . . . . . . . . . . . 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 4597 . . . . . . . . . . . . 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 752 . . . . . . . . . . 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 3653 . . . . . . . . . 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 7685 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +𝑜 𝑥))
22359, 222mpanr1 719 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +𝑜 𝑥))
224223ancoms 468 . . . . . . . . . . . . . 14 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +𝑜 𝑥))
225224anim2i 592 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +𝑜 𝑥)))
226225an12s 860 . . . . . . . . . . . 12 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ∈ On ∧ Lim (𝐵 +𝑜 𝑥)))
227 ovex 6718 . . . . . . . . . . . . 13 (𝐵 +𝑜 𝑥) ∈ V
228 omlim 7658 . . . . . . . . . . . . 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 719 . . . . . . . . . . . 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 763 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 𝐵) ∈ On)
23359jctl 563 . . . . . . . . . . . . . . . . 17 (Lim 𝑥 → (𝑥 ∈ V ∧ Lim 𝑥))
234233anim2i 592 . . . . . . . . . . . . . . . 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 7703 . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . 13 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝑥))
239 ovex 6718 . . . . . . . . . . . . 13 (𝐴 ·𝑜 𝑥) ∈ V
240238, 239jctil 559 . . . . . . . . . . . 12 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) ∈ V ∧ Lim (𝐴 ·𝑜 𝑥)))
241 oalim 7657 . . . . . . . . . . . 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 694 . . . . . . . . . . 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 753 . . . . . . . . . 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 2695 . . . . . . . . 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 639 . . . . . . . 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 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))))))
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 7638 . . . . 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 7106 . . 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 1275 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 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948   ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690   +𝑜 coa 7602   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610
This theorem is referenced by:  omass  7705  oeeui  7727  oaabs2  7770
  Copyright terms: Public domain W3C validator