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Theorem omass 7829
Description: Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (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 6821 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 ∅))
2 oveq2 6821 . . . . . . 7 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
32oveq2d 6829 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)))
41, 3eqeq12d 2775 . . . . 5 (𝑥 = ∅ → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅))))
5 oveq2 6821 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
6 oveq2 6821 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
76oveq2d 6829 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
85, 7eqeq12d 2775 . . . . 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 6821 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦))
10 oveq2 6821 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
1110oveq2d 6829 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)))
129, 11eqeq12d 2775 . . . . 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 6821 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶))
14 oveq2 6821 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
1514oveq2d 6829 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
1613, 15eqeq12d 2775 . . . . 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 7785 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
18 om0 7766 . . . . . . 7 ((𝐴 ·𝑜 𝐵) ∈ On → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = ∅)
1917, 18syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = ∅)
20 om0 7766 . . . . . . . 8 (𝐵 ∈ On → (𝐵 ·𝑜 ∅) = ∅)
2120oveq2d 6829 . . . . . . 7 (𝐵 ∈ On → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = (𝐴 ·𝑜 ∅))
22 om0 7766 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
2321, 22sylan9eqr 2816 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = ∅)
2419, 23eqtr4d 2797 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)))
25 oveq1 6820 . . . . . . . . 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 7775 . . . . . . . . . . 11 (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)))
2717, 26stoic3 1850 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)))
28 omsuc 7775 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
29283adant1 1125 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3029oveq2d 6829 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) = (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
31 omcl 7785 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
32 odi 7828 . . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . . 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 1113 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 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 1102 . . . . . . . . . . 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 2794 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
4027, 39eqeq12d 2775 . . . . . . . . 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 236 . . . . . . . 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 1113 . . . . . . 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 446 . . . . 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 468 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
46 vex 3343 . . . . . . . . . . . . . . 15 𝑥 ∈ V
47 omlim 7782 . . . . . . . . . . . . . . 15 (((𝐴 ·𝑜 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
4846, 47mpanr1 721 . . . . . . . . . . . . . 14 (((𝐴 ·𝑜 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
4945, 48sylan 489 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
5049an32s 881 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
5150ad2antrr 764 . . . . . . . . . . 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 4689 . . . . . . . . . . . 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 5949 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5446, 53mpan 708 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑥𝑥 ∈ On)
5554anim1i 593 . . . . . . . . . . . . . . . . . . . 20 ((Lim 𝑥𝐵 ∈ On) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
5655ancoms 468 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
57 omordi 7815 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·𝑜 𝑦) ∈ (𝐵 ·𝑜 𝑥)))
5856, 57sylan 489 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·𝑜 𝑦) ∈ (𝐵 ·𝑜 𝑥)))
59 ssid 3765 . . . . . . . . . . . . . . . . . . 19 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))
60 oveq2 6821 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
6160sseq2d 3774 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧) ↔ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
6261rspcev 3449 . . . . . . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . . . . . 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 3103 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∀𝑦𝑥𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧))
66 iunss2 4717 . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
69 omcl 7785 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ·𝑜 𝑥) ∈ On)
7054, 69sylan2 492 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·𝑜 𝑥) ∈ On)
71 onelon 5909 . . . . . . . . . . . . . . . . . . . 20 (((𝐵 ·𝑜 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
7270, 71sylan 489 . . . . . . . . . . . . . . . . . . 19 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
7372adantlr 753 . . . . . . . . . . . . . . . . . 18 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
74 omordlim 7826 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦))
7574ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
7646, 75mpanr1 721 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
7776ad2antlr 765 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
78 onelon 5909 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
7954, 78sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
8079, 31sylan2 492 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐵 ·𝑜 𝑦) ∈ On)
81 onelss 5927 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐵 ·𝑜 𝑦) ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → 𝑧 ⊆ (𝐵 ·𝑜 𝑦)))
82813ad2ant2 1129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → 𝑧 ⊆ (𝐵 ·𝑜 𝑦)))
83 omwordi 7820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1113 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 638 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ On → (𝐵 ∈ On → (Lim 𝑥 → (𝑦𝑥 → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))))
8887imp32 448 . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑦𝑥 → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))))
9190reximdvai 3153 . . . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
9493imp4c 618 . . . . . . . . . . . . . . . . . 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 3104 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 ·𝑜 𝑥)∃𝑦𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
97 iunss2 4717 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ 𝑦𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
10068, 99eqssd 3761 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
101 omlimcl 7827 . . . . . . . . . . . . . . . . 17 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·𝑜 𝑥))
10246, 101mpanlr1 724 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·𝑜 𝑥))
103 ovex 6841 . . . . . . . . . . . . . . . . 17 (𝐵 ·𝑜 𝑥) ∈ V
104 omlim 7782 . . . . . . . . . . . . . . . . 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 721 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ Lim (𝐵 ·𝑜 𝑥)) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
106102, 105sylan2 492 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ ((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
107106ancoms 468 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
108107an32s 881 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
109100, 108eqtr4d 2797 . . . . . . . . . . . 12 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))
11052, 109sylan9eqr 2816 . . . . . . . . . . 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 2794 . . . . . . . . . 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 631 . . . . . . . . 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 5894 . . . . . . . . . . . . 13 (𝐵 ∈ On → Ord 𝐵)
114 ord0eln0 5940 . . . . . . . . . . . . . 14 (Ord 𝐵 → (∅ ∈ 𝐵𝐵 ≠ ∅))
115114necon2bbid 2975 . . . . . . . . . . . . 13 (Ord 𝐵 → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
116113, 115syl 17 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
117116ad2antrr 764 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
118 oveq2 6821 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
119118, 22sylan9eqr 2816 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
120119oveq1d 6828 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
121 om0r 7788 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (∅ ·𝑜 𝑥) = ∅)
122120, 121sylan9eqr 2816 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 = ∅)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∅)
123122anassrs 683 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∅)
124 oveq1 6820 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐵 ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
125124, 121sylan9eqr 2816 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐵 ·𝑜 𝑥) = ∅)
126125oveq2d 6829 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 ∅))
127126, 22sylan9eq 2814 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 = ∅) ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∅)
128127an32s 881 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∅)
129123, 128eqtr4d 2797 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))
130129ex 449 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
13154, 130sylan 489 . . . . . . . . . . . 12 ((Lim 𝑥𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
132131adantll 752 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
133117, 132sylbird 250 . . . . . . . . . 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 170 . . . . . . . 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 631 . . . . . . 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 446 . . . . 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 7229 . . . 4 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶))))
140139expd 451 . . 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 1102 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715  c0 4058   ciun 4672  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886  (class class class)co 6813   +𝑜 coa 7726   ·𝑜 comu 7727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734
This theorem is referenced by:  oeoalem  7845  omabs  7896
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