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Theorem omass 8618
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 7439 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o ∅))
2 oveq2 7439 . . . . . . 7 (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
32oveq2d 7447 . . . . . 6 (𝑥 = ∅ → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o ∅)))
41, 3eqeq12d 2753 . . . . 5 (𝑥 = ∅ → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅))))
5 oveq2 7439 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝑦))
6 oveq2 7439 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
76oveq2d 7447 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝑦)))
85, 7eqeq12d 2753 . . . . 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 7439 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o suc 𝑦))
10 oveq2 7439 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
1110oveq2d 7447 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o suc 𝑦)))
129, 11eqeq12d 2753 . . . . 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 7439 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝐶))
14 oveq2 7439 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
1514oveq2d 7447 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝐶)))
1613, 15eqeq12d 2753 . . . . 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 8574 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
18 om0 8555 . . . . . . 7 ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
1917, 18syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
20 om0 8555 . . . . . . . 8 (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
2120oveq2d 7447 . . . . . . 7 (𝐵 ∈ On → (𝐴 ·o (𝐵 ·o ∅)) = (𝐴 ·o ∅))
22 om0 8555 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2321, 22sylan9eqr 2799 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 ·o ∅)) = ∅)
2419, 23eqtr4d 2780 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅)))
25 oveq1 7438 . . . . . . . . 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 8564 . . . . . . . . . . 11 (((𝐴 ·o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
2717, 26stoic3 1776 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
28 omsuc 8564 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
29283adant1 1131 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
3029oveq2d 7447 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)))
31 omcl 8574 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
32 odi 8617 . . . . . . . . . . . . . . . . 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 1165 . . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . 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 2777 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
4027, 39eqeq12d 2753 . . . . . . . . 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, 40imbitrrid 246 . . . . . . . 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 1120 . . . . . . 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 410 . . . . 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 458 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
46 vex 3484 . . . . . . . . . . . . . . 15 𝑥 ∈ V
47 omlim 8571 . . . . . . . . . . . . . . 15 (((𝐴 ·o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ·o 𝐵) ·o 𝑥) = 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
4846, 47mpanr1 703 . . . . . . . . . . . . . 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 652 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝑥) = 𝑦𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
5150ad2antrr 726 . . . . . . . . . . 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 5011 . . . . . . . . . . . 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 6448 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5446, 53mpan 690 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑥𝑥 ∈ On)
5554anim1i 615 . . . . . . . . . . . . . . . . . . . 20 ((Lim 𝑥𝐵 ∈ On) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
5655ancoms 458 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
57 omordi 8604 . . . . . . . . . . . . . . . . . . 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 4006 . . . . . . . . . . . . . . . . . . 19 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))
60 oveq2 7439 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) = (𝐴 ·o (𝐵 ·o 𝑦)))
6160sseq2d 4016 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 ·o 𝑦) → ((𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧) ↔ (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
6261rspcev 3622 . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∀𝑦𝑥𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧))
66 iunss2 5049 . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
69 omcl 8574 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ·o 𝑥) ∈ On)
7054, 69sylan2 593 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) ∈ On)
71 onelon 6409 . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . 18 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
74 omordlim 8615 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦))
7574ex 412 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
7646, 75mpanr1 703 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
7776ad2antlr 727 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
78 onelon 6409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6426 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐵 ·o 𝑦) ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → 𝑧 ⊆ (𝐵 ·o 𝑦)))
82813ad2ant2 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑦) → 𝑧 ⊆ (𝐵 ·o 𝑦)))
83 omwordi 8609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 433 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ On → (𝐵 ∈ On → (Lim 𝑥 → (𝑦𝑥 → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))))
8887imp32 418 . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑦𝑥 → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))))
9190reximdvai 3165 . . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))
9493imp4c 423 . . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 ·o 𝑥)∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))
97 iunss2 5049 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)))
10068, 99eqssd 4001 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
101 omlimcl 8616 . . . . . . . . . . . . . . . . 17 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝑥))
10246, 101mpanlr1 706 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝑥))
103 ovex 7464 . . . . . . . . . . . . . . . . 17 (𝐵 ·o 𝑥) ∈ V
104 omlim 8571 . . . . . . . . . . . . . . . . 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 703 . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
108107an32s 652 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
109100, 108eqtr4d 2780 . . . . . . . . . . . 12 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = (𝐴 ·o (𝐵 ·o 𝑥)))
11052, 109sylan9eqr 2799 . . . . . . . . . . 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 2777 . . . . . . . . . 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 419 . . . . . . . . 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 6394 . . . . . . . . . . . . 13 (𝐵 ∈ On → Ord 𝐵)
114 ord0eln0 6439 . . . . . . . . . . . . . 14 (Ord 𝐵 → (∅ ∈ 𝐵𝐵 ≠ ∅))
115114necon2bbid 2984 . . . . . . . . . . . . 13 (Ord 𝐵 → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
116113, 115syl 17 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
117116ad2antrr 726 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
118 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
119118, 22sylan9eqr 2799 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
120119oveq1d 7446 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (∅ ·o 𝑥))
121 om0r 8577 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (∅ ·o 𝑥) = ∅)
122120, 121sylan9eqr 2799 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 = ∅)) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
123122anassrs 467 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
124 oveq1 7438 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐵 ·o 𝑥) = (∅ ·o 𝑥))
125124, 121sylan9eqr 2799 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐵 ·o 𝑥) = ∅)
126125oveq2d 7447 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o ∅))
127126, 22sylan9eq 2797 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 = ∅) ∧ 𝐴 ∈ On) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∅)
128127an32s 652 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∅)
129123, 128eqtr4d 2780 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))
130129ex 412 . . . . . . . . . . . . 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 714 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
133117, 132sylbird 260 . . . . . . . . . 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 419 . . . . . . 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 410 . . . . 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 7886 . . . 4 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))))
140139expd 415 . . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  c0 4333   ciun 4991  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431   +o coa 8503   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511
This theorem is referenced by:  oeoalem  8634  omabs  8689
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