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Theorem omass 7547
 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 6557 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 ∅))
2 oveq2 6557 . . . . . . 7 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
32oveq2d 6565 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)))
41, 3eqeq12d 2625 . . . . 5 (𝑥 = ∅ → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅))))
5 oveq2 6557 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
6 oveq2 6557 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
76oveq2d 6565 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
85, 7eqeq12d 2625 . . . . 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 6557 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦))
10 oveq2 6557 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
1110oveq2d 6565 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)))
129, 11eqeq12d 2625 . . . . 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 6557 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶))
14 oveq2 6557 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
1514oveq2d 6565 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
1613, 15eqeq12d 2625 . . . . 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 7503 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
18 om0 7484 . . . . . . 7 ((𝐴 ·𝑜 𝐵) ∈ On → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = ∅)
1917, 18syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = ∅)
20 om0 7484 . . . . . . . 8 (𝐵 ∈ On → (𝐵 ·𝑜 ∅) = ∅)
2120oveq2d 6565 . . . . . . 7 (𝐵 ∈ On → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = (𝐴 ·𝑜 ∅))
22 om0 7484 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
2321, 22sylan9eqr 2666 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = ∅)
2419, 23eqtr4d 2647 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)))
25 oveq1 6556 . . . . . . . . 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 7493 . . . . . . . . . . 11 (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)))
2717, 26stoic3 1692 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)))
28 omsuc 7493 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
29283adant1 1072 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3029oveq2d 6565 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) = (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
31 omcl 7503 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
32 odi 7546 . . . . . . . . . . . . . . . . 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 1352 . . . . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . . . 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 89 . . . . . . . . . . . . 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 51 . . . . . . . . . . . 12 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))))
38373imp 1249 . . . . . . . . . . 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 2644 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
4027, 39eqeq12d 2625 . . . . . . . . 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 235 . . . . . . . 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 1256 . . . . . . 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 85 . . . . . 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 3176 . . . . . . . . . . . . . . 15 𝑥 ∈ V
47 omlim 7500 . . . . . . . . . . . . . . 15 (((𝐴 ·𝑜 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
4846, 47mpanr1 715 . . . . . . . . . . . . . 14 (((𝐴 ·𝑜 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
4945, 48sylan 487 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
5049an32s 842 . . . . . . . . . . . 12 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = 𝑦𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
5150ad2antrr 758 . . . . . . . . . . 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 4473 . . . . . . . . . . . 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 5705 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5446, 53mpan 702 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑥𝑥 ∈ On)
5554anim1i 590 . . . . . . . . . . . . . . . . . . . 20 ((Lim 𝑥𝐵 ∈ On) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
5655ancoms 468 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
57 omordi 7533 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·𝑜 𝑦) ∈ (𝐵 ·𝑜 𝑥)))
5856, 57sylan 487 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·𝑜 𝑦) ∈ (𝐵 ·𝑜 𝑥)))
59 ssid 3587 . . . . . . . . . . . . . . . . . . 19 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))
60 oveq2 6557 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
6160sseq2d 3596 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧) ↔ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
6261rspcev 3282 . . . . . . . . . . . . . . . . . . 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 703 . . . . . . . . . . . . . . . . . 18 ((𝐵 ·𝑜 𝑦) ∈ (𝐵 ·𝑜 𝑥) → ∃𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧))
6458, 63syl6 34 . . . . . . . . . . . . . . . . 17 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → ∃𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧)))
6564ralrimiv 2948 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∀𝑦𝑥𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧))
66 iunss2 4501 . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
69 omcl 7503 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ·𝑜 𝑥) ∈ On)
7054, 69sylan2 490 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·𝑜 𝑥) ∈ On)
71 onelon 5665 . . . . . . . . . . . . . . . . . . . 20 (((𝐵 ·𝑜 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
7270, 71sylan 487 . . . . . . . . . . . . . . . . . . 19 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
7372adantlr 747 . . . . . . . . . . . . . . . . . 18 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
74 omordlim 7544 . . . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
7776ad2antlr 759 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
78 onelon 5665 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
7954, 78sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
8079, 31sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐵 ·𝑜 𝑦) ∈ On)
81 onelss 5683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐵 ·𝑜 𝑦) ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → 𝑧 ⊆ (𝐵 ·𝑜 𝑦)))
82813ad2ant2 1076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → 𝑧 ⊆ (𝐵 ·𝑜 𝑦)))
83 omwordi 7538 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
8482, 83syld 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
85843exp 1256 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ On → ((𝐵 ·𝑜 𝑦) ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
8680, 85syl5 33 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ On → ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
8786exp4d 635 . . . . . . . . . . . . . . . . . . . . . . . . 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 84 . . . . . . . . . . . . . . . . . . . . . . 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 2998 . . . . . . . . . . . . . . . . . . . . 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 46 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
9392exp31 628 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
9493imp4c 615 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ On → ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → ∃𝑦𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
9573, 94mpcom 37 . . . . . . . . . . . . . . . . 17 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → ∃𝑦𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
9695ralrimiva 2949 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 ·𝑜 𝑥)∃𝑦𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
97 iunss2 4501 . . . . . . . . . . . . . . . 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 3585 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
101 omlimcl 7545 . . . . . . . . . . . . . . . . 17 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·𝑜 𝑥))
10246, 101mpanlr1 718 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·𝑜 𝑥))
103 ovex 6577 . . . . . . . . . . . . . . . . 17 (𝐵 ·𝑜 𝑥) ∈ V
104 omlim 7500 . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ Lim (𝐵 ·𝑜 𝑥)) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
106102, 105sylan2 490 . . . . . . . . . . . . . . 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 842 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
109100, 108eqtr4d 2647 . . . . . . . . . . . 12 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))
11052, 109sylan9eqr 2666 . . . . . . . . . . 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 2644 . . . . . . . . . 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 628 . . . . . . . . 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 5650 . . . . . . . . . . . . 13 (𝐵 ∈ On → Ord 𝐵)
114 ord0eln0 5696 . . . . . . . . . . . . . 14 (Ord 𝐵 → (∅ ∈ 𝐵𝐵 ≠ ∅))
115114necon2bbid 2825 . . . . . . . . . . . . 13 (Ord 𝐵 → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
116113, 115syl 17 . . . . . . . . . . . 12 (𝐵 ∈ On → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
117116ad2antrr 758 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
118 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
119118, 22sylan9eqr 2666 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
120119oveq1d 6564 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
121 om0r 7506 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (∅ ·𝑜 𝑥) = ∅)
122120, 121sylan9eqr 2666 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 = ∅)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∅)
123122anassrs 678 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∅)
124 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐵 ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
125124, 121sylan9eqr 2666 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐵 ·𝑜 𝑥) = ∅)
126125oveq2d 6565 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 ∅))
127126, 22sylan9eq 2664 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 = ∅) ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∅)
128127an32s 842 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∅)
129123, 128eqtr4d 2647 . . . . . . . . . . . . . 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 487 . . . . . . . . . . . 12 ((Lim 𝑥𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
132131adantll 746 . . . . . . . . . . 11 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
133117, 132sylbird 249 . . . . . . . . . 10 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (¬ ∅ ∈ 𝐵 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
134133a1dd 48 . . . . . . . . 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 169 . . . . . . . 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 628 . . . . . . 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 87 . . . . . 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 6956 . . . 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 87 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))))
1421413imp 1249 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 195   ∧ 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:  oeoalem  7563  omabs  7614
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