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

Theorem omass 8527
Description: Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. Theorem 4.4 of [Schloeder] p. 13. (Contributed by NM, 28-Dec-2004.)
Assertion
Ref Expression
omass ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))

Proof of Theorem omass
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7365 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o ∅))
2 oveq2 7365 . . . . . . 7 (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
32oveq2d 7373 . . . . . 6 (𝑥 = ∅ → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o ∅)))
41, 3eqeq12d 2752 . . . . 5 (𝑥 = ∅ → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅))))
5 oveq2 7365 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝑦))
6 oveq2 7365 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
76oveq2d 7373 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝑦)))
85, 7eqeq12d 2752 . . . . 5 (𝑥 = 𝑦 → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦))))
9 oveq2 7365 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o suc 𝑦))
10 oveq2 7365 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
1110oveq2d 7373 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o suc 𝑦)))
129, 11eqeq12d 2752 . . . . 5 (𝑥 = suc 𝑦 → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))
13 oveq2 7365 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝐶))
14 oveq2 7365 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
1514oveq2d 7373 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝐶)))
1613, 15eqeq12d 2752 . . . . 5 (𝑥 = 𝐶 → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))))
17 omcl 8482 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
18 om0 8463 . . . . . . 7 ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
1917, 18syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
20 om0 8463 . . . . . . . 8 (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
2120oveq2d 7373 . . . . . . 7 (𝐵 ∈ On → (𝐴 ·o (𝐵 ·o ∅)) = (𝐴 ·o ∅))
22 om0 8463 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2321, 22sylan9eqr 2798 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 ·o ∅)) = ∅)
2419, 23eqtr4d 2779 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅)))
25 oveq1 7364 . . . . . . . . 9 (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
26 omsuc 8472 . . . . . . . . . . 11 (((𝐴 ·o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
2717, 26stoic3 1778 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
28 omsuc 8472 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
29283adant1 1130 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
3029oveq2d 7373 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)))
31 omcl 8482 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
32 odi 8526 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
3331, 32syl3an2 1164 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐵 ∈ On) → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
34333exp 1119 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ On → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))))
3534expd 416 . . . . . . . . . . . . . 14 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐵 ∈ On → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵))))))
3635com34 91 . . . . . . . . . . . . 13 (𝐴 ∈ On → (𝐵 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵))))))
3736pm2.43d 53 . . . . . . . . . . . 12 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))))
38373imp 1111 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
3930, 38eqtrd 2776 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
4027, 39eqeq12d 2752 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦)) ↔ (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵))))
4125, 40syl5ibr 245 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))
42413exp 1119 . . . . . . 7 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))))
4342com3r 87 . . . . . 6 (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))))
4443impd 411 . . . . 5 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦)))))
4517ancoms 459 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
46 vex 3449 . . . . . . . . . . . . . . 15 𝑥 ∈ V
47 omlim 8479 . . . . . . . . . . . . . . 15 (((𝐴 ·o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ·o 𝐵) ·o 𝑥) = 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
4846, 47mpanr1 701 . . . . . . . . . . . . . 14 (((𝐴 ·o 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 ·o 𝐵) ·o 𝑥) = 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
4945, 48sylan 580 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ Lim 𝑥) → ((𝐴 ·o 𝐵) ·o 𝑥) = 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
5049an32s 650 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝑥) = 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
5150ad2antrr 724 . . . . . . . . . . 11 (((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) ∧ ∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦))) → ((𝐴 ·o 𝐵) ·o 𝑥) = 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
52 iuneq2 4973 . . . . . . . . . . . 12 (∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)))
53 limelon 6381 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5446, 53mpan 688 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑥𝑥 ∈ On)
5554anim1i 615 . . . . . . . . . . . . . . . . . . . 20 ((Lim 𝑥𝐵 ∈ On) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
5655ancoms 459 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
57 omordi 8513 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥)))
5856, 57sylan 580 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥)))
59 ssid 3966 . . . . . . . . . . . . . . . . . . 19 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))
60 oveq2 7365 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) = (𝐴 ·o (𝐵 ·o 𝑦)))
6160sseq2d 3976 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 ·o 𝑦) → ((𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧) ↔ (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
6261rspcev 3581 . . . . . . . . . . . . . . . . . . 19 (((𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥) ∧ (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))) → ∃𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧))
6359, 62mpan2 689 . . . . . . . . . . . . . . . . . 18 ((𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥) → ∃𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧))
6458, 63syl6 35 . . . . . . . . . . . . . . . . 17 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → ∃𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧)))
6564ralrimiv 3142 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∀𝑦𝑥𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧))
66 iunss2 5009 . . . . . . . . . . . . . . . 16 (∀𝑦𝑥𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
6765, 66syl 17 . . . . . . . . . . . . . . 15 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
6867adantlr 713 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
69 omcl 8482 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ·o 𝑥) ∈ On)
7054, 69sylan2 593 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) ∈ On)
71 onelon 6342 . . . . . . . . . . . . . . . . . . . 20 (((𝐵 ·o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
7270, 71sylan 580 . . . . . . . . . . . . . . . . . . 19 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
7372adantlr 713 . . . . . . . . . . . . . . . . . 18 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
74 omordlim 8524 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦))
7574ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
7646, 75mpanr1 701 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
7776ad2antlr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
78 onelon 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
7954, 78sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
8079, 31sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐵 ·o 𝑦) ∈ On)
81 onelss 6359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐵 ·o 𝑦) ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → 𝑧 ⊆ (𝐵 ·o 𝑦)))
82813ad2ant2 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑦) → 𝑧 ⊆ (𝐵 ·o 𝑦)))
83 omwordi 8518 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
8482, 83syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
85843exp 1119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ On → ((𝐵 ·o 𝑦) ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))
8680, 85syl5 34 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ On → ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))
8786exp4d 434 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ On → (𝐵 ∈ On → (Lim 𝑥 → (𝑦𝑥 → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))))
8887imp32 419 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) → (𝑦𝑥 → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))
8988com23 86 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) → (𝐴 ∈ On → (𝑦𝑥 → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))
9089imp 407 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑦𝑥 → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))))
9190reximdvai 3162 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦) → ∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
9277, 91syld 47 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
9392exp31 420 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))
9493imp4c 424 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ On → ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → ∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
9573, 94mpcom 38 . . . . . . . . . . . . . . . . 17 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → ∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))
9695ralrimiva 3143 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 ·o 𝑥)∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))
97 iunss2 5009 . . . . . . . . . . . . . . . 16 (∀𝑧 ∈ (𝐵 ·o 𝑥)∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)) → 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)))
9896, 97syl 17 . . . . . . . . . . . . . . 15 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)))
9998adantr 481 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)))
10068, 99eqssd 3961 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
101 omlimcl 8525 . . . . . . . . . . . . . . . . 17 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝑥))
10246, 101mpanlr1 704 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝑥))
103 ovex 7390 . . . . . . . . . . . . . . . . 17 (𝐵 ·o 𝑥) ∈ V
104 omlim 8479 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ ((𝐵 ·o 𝑥) ∈ V ∧ Lim (𝐵 ·o 𝑥))) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
105103, 104mpanr1 701 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ Lim (𝐵 ·o 𝑥)) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
106102, 105sylan2 593 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ ((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
107106ancoms 459 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
108107an32s 650 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
109100, 108eqtr4d 2779 . . . . . . . . . . . 12 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = (𝐴 ·o (𝐵 ·o 𝑥)))
11052, 109sylan9eqr 2798 . . . . . . . . . . 11 (((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) ∧ ∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦))) → 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑥)))
11151, 110eqtrd 2776 . . . . . . . . . 10 (((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) ∧ ∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦))) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))
112111exp31 420 . . . . . . . . 9 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (∅ ∈ 𝐵 → (∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))))
113 eloni 6327 . . . . . . . . . . . . 13 (𝐵 ∈ On → Ord 𝐵)
114 ord0eln0 6372 . . . . . . . . . . . . . 14 (Ord 𝐵 → (∅ ∈ 𝐵𝐵 ≠ ∅))
115114necon2bbid 2987 . . . . . . . . . . . . 13 (Ord 𝐵 → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
116113, 115syl 17 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
117116ad2antrr 724 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
118 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
119118, 22sylan9eqr 2798 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
120119oveq1d 7372 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (∅ ·o 𝑥))
121 om0r 8485 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (∅ ·o 𝑥) = ∅)
122120, 121sylan9eqr 2798 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 = ∅)) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
123122anassrs 468 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
124 oveq1 7364 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐵 ·o 𝑥) = (∅ ·o 𝑥))
125124, 121sylan9eqr 2798 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐵 ·o 𝑥) = ∅)
126125oveq2d 7373 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o ∅))
127126, 22sylan9eq 2796 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 = ∅) ∧ 𝐴 ∈ On) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∅)
128127an32s 650 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∅)
129123, 128eqtr4d 2779 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))
130129ex 413 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
13154, 130sylan 580 . . . . . . . . . . . 12 ((Lim 𝑥𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
132131adantll 712 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
133117, 132sylbird 259 . . . . . . . . . 10 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (¬ ∅ ∈ 𝐵 → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
134133a1dd 50 . . . . . . . . 9 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (¬ ∅ ∈ 𝐵 → (∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))))
135112, 134pm2.61d 179 . . . . . . . 8 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
136135exp31 420 . . . . . . 7 (𝐵 ∈ On → (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))))
137136com3l 89 . . . . . 6 (Lim 𝑥 → (𝐴 ∈ On → (𝐵 ∈ On → (∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))))
138137impd 411 . . . . 5 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))))
1394, 8, 12, 16, 24, 44, 138tfinds3 7801 . . . 4 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))))
140139expd 416 . . 3 (𝐶 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))))
141140com3l 89 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))))
1421413imp 1111 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282   ciun 4954  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357   +o coa 8409   ·o comu 8410
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417
This theorem is referenced by:  oeoalem  8543  omabs  8597
  Copyright terms: Public domain W3C validator