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.

Step	Hyp	Ref	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 ∅))
3	2	oveq2d 6565	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)))
4	1, 3	eqeq12d 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))
7	6	oveq2d 6565	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
8	5, 7	eqeq12d 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))
11	10	oveq2d 6565	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)))
12	9, 11	eqeq12d 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))
15	14	oveq2d 6565	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
16	13, 15	eqeq12d 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 ∅) = ∅)
19	17, 18	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = ∅)
20		om0 7484	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 ·𝑜 ∅) = ∅)
21	20	oveq2d 6565	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = (𝐴 ·𝑜 ∅))
22		om0 7484	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
23	21, 22	sylan9eqr 2666	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = ∅)
24	19, 23	eqtr4d 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)))
27	17, 26	stoic3 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))
29	28	3adant1 1072	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
30	29	oveq2d 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)))
33	31, 32	syl3an2 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)))
34	33	3exp 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)))))
35	34	expd 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))))))
36	35	com34 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))))))
37	36	pm2.43d 51	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))))
38	37	3imp 1249	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
39	30, 38	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
40	27, 39	eqeq12d 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))))
41	25, 40	syl5ibr 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))))
42	41	3exp 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))))))
43	42	com3r 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))))))
44	43	impd 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)))))
45	17	ancoms 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) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
48	46, 47	mpanr1 715	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ·𝑜 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
49	45, 48	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
50	49	an32s 842	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
51	50	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
52		iuneq2 4473	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ∪ 𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
53		limelon 5705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
54	46, 53	mpan 702	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
55	54	anim1i 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
56	55	ancoms 468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
57		omordi 7533	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝑦 ∈ 𝑥 → (𝐵 ·𝑜 𝑦) ∈ (𝐵 ·𝑜 𝑥)))
58	56, 57	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦 ∈ 𝑥 → (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)))
61	60	sseq2d 3596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧) ↔ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
62	61	rspcev 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))
63	59, 62	mpan2 703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ·𝑜 𝑦) ∈ (𝐵 ·𝑜 𝑥) → ∃𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧))
64	58, 63	syl6 34	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧)))
65	64	ralrimiv 2948	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧))
66		iunss2 4501	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ (𝐴 ·𝑜 𝑧) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
67	65, 66	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
68	67	adantlr 747	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
69		omcl 7503	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ·𝑜 𝑥) ∈ On)
70	54, 69	sylan2 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·𝑜 𝑥) ∈ On)
71		onelon 5665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐵 ·𝑜 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
72	70, 71	sylan 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → 𝑧 ∈ On)
73	72	adantlr 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)) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦))
75	74	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
76	46, 75	mpanr1 715	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
77	76	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦)))
78		onelon 5665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
79	54, 78	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
80	79, 31	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 ∈ On ∧ (Lim 𝑥 ∧ 𝑦 ∈ 𝑥)) → (𝐵 ·𝑜 𝑦) ∈ On)
81		onelss 5683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐵 ·𝑜 𝑦) ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → 𝑧 ⊆ (𝐵 ·𝑜 𝑦)))
82	81	3ad2ant2 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))))
84	82, 83	syld 46	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
85	84	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ On → ((𝐵 ·𝑜 𝑦) ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
86	80, 85	syl5 33	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ On → ((𝐵 ∈ On ∧ (Lim 𝑥 ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
87	86	exp4d 635	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ On → (𝐵 ∈ On → (Lim 𝑥 → (𝑦 ∈ 𝑥 → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))))
88	87	imp32 448	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) → (𝑦 ∈ 𝑥 → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
89	88	com23 84	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) → (𝐴 ∈ On → (𝑦 ∈ 𝑥 → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
90	89	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝑥 → (𝑧 ∈ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))))
91	90	reximdvai 2998	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·𝑜 𝑦) → ∃𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
92	77, 91	syld 46	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
93	92	exp31 628	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ On → ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐴 ∈ On → (𝑧 ∈ (𝐵 ·𝑜 𝑥) → ∃𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))))
94	93	imp4c 615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ On → ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
95	73, 94	mpcom 37	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·𝑜 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
96	95	ralrimiva 2949	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 ·𝑜 𝑥)∃𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
97		iunss2 4501	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ (𝐵 ·𝑜 𝑥)∃𝑦 ∈ 𝑥 (𝐴 ·𝑜 𝑧) ⊆ (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
98	96, 97	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
99	98	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
100	68, 99	eqssd 3585	. . . . . . . . . . . . 13 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → ∪ 𝑦 ∈ 𝑥 (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))
102	46, 101	mpanlr1 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))
105	103, 104	mpanr1 715	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ Lim (𝐵 ·𝑜 𝑥)) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
106	102, 105	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
107	106	ancoms 468	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
108	107	an32s 842	. . . . . . . . . . . . 13 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·𝑜 𝑥)(𝐴 ·𝑜 𝑧))
109	100, 108	eqtr4d 2647	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))
110	52, 109	sylan9eqr 2666	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))) → ∪ 𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))
111	51, 110	eqtrd 2644	. . . . . . . . . 10 ⊢ (((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) ∧ ∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))
112	111	exp31 628	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (∅ ∈ 𝐵 → (∀𝑦 ∈ 𝑥 ((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 → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
115	114	necon2bbid 2825	. . . . . . . . . . . . 13 ⊢ (Ord 𝐵 → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
116	113, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
117	116	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
118		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
119	118, 22	sylan9eqr 2666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
120	119	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
121		om0r 7506	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (∅ ·𝑜 𝑥) = ∅)
122	120, 121	sylan9eqr 2666	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 = ∅)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∅)
123	122	anassrs 678	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ∅)
124		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = ∅ → (𝐵 ·𝑜 𝑥) = (∅ ·𝑜 𝑥))
125	124, 121	sylan9eqr 2666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐵 ·𝑜 𝑥) = ∅)
126	125	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 ∅))
127	126, 22	sylan9eq 2664	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On ∧ 𝐵 = ∅) ∧ 𝐴 ∈ On) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∅)
128	127	an32s 842	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = ∅)
129	123, 128	eqtr4d 2647	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))
130	129	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
131	54, 130	sylan 487	. . . . . . . . . . . 12 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
132	131	adantll 746	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
133	117, 132	sylbird 249	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (¬ ∅ ∈ 𝐵 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
134	133	a1dd 48	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (¬ ∅ ∈ 𝐵 → (∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))))
135	112, 134	pm2.61d 169	. . . . . . . 8 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
136	135	exp31 628	. . . . . . 7 ⊢ (𝐵 ∈ On → (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))))
137	136	com3l 87	. . . . . 6 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (𝐵 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))))
138	137	impd 446	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦 ∈ 𝑥 ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)))))
139	4, 8, 12, 16, 24, 44, 138	tfinds3 6956	. . . 4 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶))))
140	139	expd 451	. . 3 ⊢ (𝐶 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))))
141	140	com3l 87	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))))
142	141	3imp 1249	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))

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