Theorem omass 8205

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 7163	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o ∅))
2		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
3	2	oveq2d 7171	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o ∅)))
4	1, 3	eqeq12d 2837	. . . . 5 ⊢ (𝑥 = ∅ → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅))))
5		oveq2 7163	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝑦))
6		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
7	6	oveq2d 7171	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝑦)))
8	5, 7	eqeq12d 2837	. . . . 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 7163	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o suc 𝑦))
10		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
11	10	oveq2d 7171	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o suc 𝑦)))
12	9, 11	eqeq12d 2837	. . . . 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 7163	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝐶))
14		oveq2 7163	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
15	14	oveq2d 7171	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝐶)))
16	13, 15	eqeq12d 2837	. . . . 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 8160	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
18		om0 8141	. . . . . . 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 8141	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 ·o ∅) = ∅)
21	20	oveq2d 7171	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ·o (𝐵 ·o ∅)) = (𝐴 ·o ∅))
22		om0 8141	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
23	21, 22	sylan9eqr 2878	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o (𝐵 ·o ∅)) = ∅)
24	19, 23	eqtr4d 2859	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅)))
25		oveq1 7162	. . . . . . . . 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 8150	. . . . . . . . . . 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 1773	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
28		omsuc 8150	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
29	28	3adant1 1126	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
30	29	oveq2d 7171	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)))
31		omcl 8160	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·o 𝑦) ∈ On)
32		odi 8204	. . . . . . . . . . . . . . . . 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 1160	. . . . . . . . . . . . . . . 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 1115	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . 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 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))))))
37	36	pm2.43d 53	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))))
38	37	3imp 1107	. . . . . . . . . . 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 2856	. . . . . . . . . 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 2837	. . . . . . . . 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 248	. . . . . . . 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 1115	. . . . . . 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 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))))))
44	43	impd 413	. . . . 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 461	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
46		vex 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
47		omlim 8157	. . . . . . . . . . . . . . 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 701	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
49	45, 48	sylan 582	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ Lim 𝑥) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
50	49	an32s 650	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 ((𝐴 ·o 𝐵) ·o 𝑦))
51	50	ad2antrr 724	. . . . . . . . . . 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 4937	. . . . . . . . . . . 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 6253	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
54	46, 53	mpan 688	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
55	54	anim1i 616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
56	55	ancoms 461	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑥 ∈ On ∧ 𝐵 ∈ On))
57		omordi 8191	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝑦 ∈ 𝑥 → (𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥)))
58	56, 57	sylan 582	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦 ∈ 𝑥 → (𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥)))
59		ssid 3988	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))
60		oveq2 7163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) = (𝐴 ·o (𝐵 ·o 𝑦)))
61	60	sseq2d 3998	. . . . . . . . . . . . . . . . . . . 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 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))
63	59, 62	mpan2 689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ·o 𝑦) ∈ (𝐵 ·o 𝑥) → ∃𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧))
64	58, 63	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧)))
65	64	ralrimiv 3181	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o (𝐵 ·o 𝑦)) ⊆ (𝐴 ·o 𝑧))
66		iunss2 4972	. . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) ⊆ ∪ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
69		omcl 8160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ·o 𝑥) ∈ On)
70	54, 69	sylan2 594	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝐵 ·o 𝑥) ∈ On)
71		onelon 6215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐵 ·o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
72	70, 71	sylan 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
73	72	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → 𝑧 ∈ On)
74		omordlim 8202	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·o 𝑦))
75	74	ex 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
76	46, 75	mpanr1 701	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
77	76	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝑥)) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑥) → ∃𝑦 ∈ 𝑥 𝑧 ∈ (𝐵 ·o 𝑦)))
78		onelon 6215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
79	54, 78	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Lim 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
80	79, 31	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐵 ∈ On ∧ (Lim 𝑥 ∧ 𝑦 ∈ 𝑥)) → (𝐵 ·o 𝑦) ∈ On)
81		onelss 6232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐵 ·o 𝑦) ∈ On → (𝑧 ∈ (𝐵 ·o 𝑦) → 𝑧 ⊆ (𝐵 ·o 𝑦)))
82	81	3ad2ant2 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑧 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑦) → 𝑧 ⊆ (𝐵 ·o 𝑦)))
83		omwordi 8196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 ∈ On ∧ (𝐵 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 ·o 𝑦) → (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦))))
85	84	3exp 1115	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 34	. . . . . . . . . . . . . . . . . . . . . . . . . 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 436	. . . . . . . . . . . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . . . . . . 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 3272	. . . . . . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . . . . 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 38	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ (𝐵 ·o 𝑥)) → ∃𝑦 ∈ 𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))
96	95	ralrimiva 3182	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 ·o 𝑥)∃𝑦 ∈ 𝑥 (𝐴 ·o 𝑧) ⊆ (𝐴 ·o (𝐵 ·o 𝑦)))
97		iunss2 4972	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . 13 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = ∪ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
101		omlimcl 8203	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝑥))
102	46, 101	mpanlr1 704	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝑥))
103		ovex 7188	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ·o 𝑥) ∈ V
104		omlim 8157	. . . . . . . . . . . . . . . . 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 701	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ Lim (𝐵 ·o 𝑥)) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
106	102, 105	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ ((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
107	106	ancoms 461	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
108	107	an32s 650	. . . . . . . . . . . . 13 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∪ 𝑧 ∈ (𝐵 ·o 𝑥)(𝐴 ·o 𝑧))
109	100, 108	eqtr4d 2859	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐵) → ∪ 𝑦 ∈ 𝑥 (𝐴 ·o (𝐵 ·o 𝑦)) = (𝐴 ·o (𝐵 ·o 𝑥)))
110	52, 109	sylan9eqr 2878	. . . . . . . . . . 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 2856	. . . . . . . . . 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 422	. . . . . . . . 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 6200	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → Ord 𝐵)
114		ord0eln0 6244	. . . . . . . . . . . . . 14 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
115	114	necon2bbid 3059	. . . . . . . . . . . . 13 ⊢ (Ord 𝐵 → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
116	113, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
117	116	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ ↔ ¬ ∅ ∈ 𝐵))
118		oveq2 7163	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
119	118, 22	sylan9eqr 2878	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
120	119	oveq1d 7170	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (∅ ·o 𝑥))
121		om0r 8163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (∅ ·o 𝑥) = ∅)
122	120, 121	sylan9eqr 2878	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 = ∅)) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
123	122	anassrs 470	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = ∅)
124		oveq1 7162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = ∅ → (𝐵 ·o 𝑥) = (∅ ·o 𝑥))
125	124, 121	sylan9eqr 2878	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐵 ·o 𝑥) = ∅)
126	125	oveq2d 7171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o ∅))
127	126, 22	sylan9eq 2876	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On ∧ 𝐵 = ∅) ∧ 𝐴 ∈ On) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∅)
128	127	an32s 650	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → (𝐴 ·o (𝐵 ·o 𝑥)) = ∅)
129	123, 128	eqtr4d 2859	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 = ∅) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)))
130	129	ex 415	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
131	54, 130	sylan 582	. . . . . . . . . . . 12 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
132	131	adantll 712	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (𝐵 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
133	117, 132	sylbird 262	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ On) → (¬ ∅ ∈ 𝐵 → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
134	133	a1dd 50	. . . . . . . . 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 181	. . . . . . . 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 422	. . . . . . 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 89	. . . . . 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 413	. . . . 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 7578	. . . 4 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))))
140	139	expd 418	. . 3 ⊢ (𝐶 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))))
141	140	com3l 89	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))))
142	141	3imp 1107	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ⊆ wss 3935  ∅c0 4290  ∪ ciun 4918  Ord word 6189  Oncon0 6190  Lim wlim 6191  suc csuc 6192  (class class class)co 7155   +_o coa 8098   ·_o comu 8099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-omul 8106

This theorem is referenced by:  oeoalem  8221  omabs  8273