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Theorem omass 8250
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 7191 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o ∅))
2 oveq2 7191 . . . . . . 7 (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
32oveq2d 7199 . . . . . 6 (𝑥 = ∅ → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o ∅)))
41, 3eqeq12d 2755 . . . . 5 (𝑥 = ∅ → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅))))
5 oveq2 7191 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝑦))
6 oveq2 7191 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
76oveq2d 7199 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝑦)))
85, 7eqeq12d 2755 . . . . 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 7191 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o suc 𝑦))
10 oveq2 7191 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
1110oveq2d 7199 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o suc 𝑦)))
129, 11eqeq12d 2755 . . . . 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 7191 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝐶))
14 oveq2 7191 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
1514oveq2d 7199 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝐶)))
1613, 15eqeq12d 2755 . . . . 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 8205 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
18 om0 8186 . . . . . . 7 ((𝐴 ·o 𝐵) ∈ On → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
1917, 18syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
20 om0 8186 . . . . . . . 8 (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
2120oveq2d 7199 . . . . . . 7 (𝐵 ∈ On → (𝐴 ·o (𝐵 ·o ∅)) = (𝐴 ·o ∅))
22 om0 8186 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2321, 22sylan9eqr 2796 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 ·o ∅)) = ∅)
2419, 23eqtr4d 2777 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅)))
25 oveq1 7190 . . . . . . . . 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 8195 . . . . . . . . . . 11 (((𝐴 ·o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
2717, 26stoic3 1783 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
28 omsuc 8195 . . . . . . . . . . . . 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 7199 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)))
31 omcl 8205 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
32 odi 8249 . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . 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 1112 . . . . . . . . . . 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 2774 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
4027, 39eqeq12d 2755 . . . . . . . . 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 249 . . . . . . . 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 414 . . . . 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 462 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
46 vex 3404 . . . . . . . . . . . . . . 15 𝑥 ∈ V
47 omlim 8202 . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . 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 4910 . . . . . . . . . . . 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 6246 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5446, 53mpan 690 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑥𝑥 ∈ On)
5554anim1i 618 . . . . . . . . . . . . . . . . . . . 20 ((Lim 𝑥𝐵 ∈ On) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
5655ancoms 462 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
57 omordi 8236 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥)))
5856, 57sylan 583 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦𝑥 → (𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥)))
59 ssid 3909 . . . . . . . . . . . . . . . . . . 19 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))
60 oveq2 7191 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) = (𝐴 ·o (𝐵 ·o 𝑦)))
6160sseq2d 3919 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 ·o 𝑦) → ((𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧) ↔ (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
6261rspcev 3529 . . . . . . . . . . . . . . . . . . 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 3096 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∀𝑦𝑥𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧))
66 iunss2 4945 . . . . . . . . . . . . . . . 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 8205 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ·o 𝑥) ∈ On)
7054, 69sylan2 596 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) ∈ On)
71 onelon 6208 . . . . . . . . . . . . . . . . . . . 20 (((𝐵 ·o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
7270, 71sylan 583 . . . . . . . . . . . . . . . . . . 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 8247 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → ∃𝑦𝑥 𝑧 ∈ (𝐵 ·o 𝑦))
7574ex 416 . . . . . . . . . . . . . . . . . . . . . . 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 6208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
7954, 78sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥𝑦𝑥) → 𝑦 ∈ On)
8079, 31sylan2 596 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ (Lim 𝑥𝑦𝑥)) → (𝐵 ·o 𝑦) ∈ On)
81 onelss 6225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ On → (𝐵 ∈ On → (Lim 𝑥 → (𝑦𝑥 → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))))
8887imp32 422 . . . . . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑦𝑥 → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))))
9190reximdvai 3183 . . . . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))))
9493imp4c 427 . . . . . . . . . . . . . . . . . 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 3097 . . . . . . . . . . . . . . . 16 (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 ·o 𝑥)∃𝑦𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))
97 iunss2 4945 . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧) ⊆ 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)))
10068, 99eqssd 3904 . . . . . . . . . . . . 13 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
101 omlimcl 8248 . . . . . . . . . . . . . . . . 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 7216 . . . . . . . . . . . . . . . . 17 (𝐵 ·o 𝑥) ∈ V
104 omlim 8202 . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ ((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (𝐵 ·o 𝑥)) = 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
107106ancoms 462 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . 12 ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → 𝑦𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = (𝐴 ·o (𝐵 ·o 𝑥)))
11052, 109sylan9eqr 2796 . . . . . . . . . . 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 2774 . . . . . . . . . 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 423 . . . . . . . . 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 6193 . . . . . . . . . . . . 13 (𝐵 ∈ On → Ord 𝐵)
114 ord0eln0 6237 . . . . . . . . . . . . . 14 (Ord 𝐵 → (∅ ∈ 𝐵𝐵 ≠ ∅))
115114necon2bbid 2978 . . . . . . . . . . . . 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 7191 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
119118, 22sylan9eqr 2796 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
120119oveq1d 7198 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (∅ ·o 𝑥))
121 om0r 8208 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (∅ ·o 𝑥) = ∅)
122120, 121sylan9eqr 2796 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 = ∅)) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
123122anassrs 471 . . . . . . . . . . . . . . 15 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
124 oveq1 7190 . . . . . . . . . . . . . . . . . . 19 (𝐵 = ∅ → (𝐵 ·o 𝑥) = (∅ ·o 𝑥))
125124, 121sylan9eqr 2796 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐵 ·o 𝑥) = ∅)
126125oveq2d 7199 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o ∅))
127126, 22sylan9eq 2794 . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . 14 (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))
130129ex 416 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
13154, 130sylan 583 . . . . . . . . . . . 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 263 . . . . . . . . . 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 182 . . . . . . . 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 423 . . . . . . 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 414 . . . . 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 7611 . . . 4 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))))
140139expd 419 . . 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 1112 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  Vcvv 3400  wss 3853  c0 4221   ciun 4891  Ord word 6182  Oncon0 6183  Lim wlim 6184  suc csuc 6185  (class class class)co 7183   +o coa 8141   ·o comu 8142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6186  df-on 6187  df-lim 6188  df-suc 6189  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-om 7613  df-wrecs 7989  df-recs 8050  df-rdg 8088  df-1o 8144  df-oadd 8148  df-omul 8149
This theorem is referenced by:  oeoalem  8266  omabs  8318
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