Theorem omordi 8554

Description: Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. Lemma 3.15 of [Schloeder] p. 9. (Contributed by NM, 14-Dec-2004.)

Assertion

Ref	Expression
omordi	⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Proof of Theorem omordi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelon 6382	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	ex 418	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
4		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
5	4	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
6	3, 5	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
7		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
8		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
9	8	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
10	7, 9	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
11		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
12		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
13	12	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
14	11, 13	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
15		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
16		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
17	16	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
18	15, 17	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
19		noel 4284	. . . . . . . . . . 11 ⊢ ¬ 𝐴 ∈ ∅
20	19	pm2.21i 120	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))
21	20	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
22		elsuci 6427	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
23		omcl 8524	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o 𝑦) ∈ On)
24		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
25	23, 24	jca 521	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On))
26		oaword1 8540	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
27	26	sseld 3930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
28	27	imim2d 58	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶))))
29	28	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
30	29	adantrl 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
31		oaord1 8539	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
32	31	biimpa 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
33		oveq2 7422	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
34	33	eleq1d 2845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
35	32, 34	syl5ibrcom 250	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
36	35	adantrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
37	30, 36	jaod 873	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ·o 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
38	25, 37	sylan 592	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
39	22, 38	syl5 35	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
40		omsuc 8514	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
41	40	eleq2d 2846	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
42	41	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
43	39, 42	sylibrd 262	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ On ∧ 𝑦 ∈ On) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
44	43	exp43 442	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (𝑦 ∈ On → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
45	44	com12 33	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
46	45	adantld 496	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
47	46	impd 416	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))))
48		id 23	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ∈ On ∧ Lim 𝑥))
49	48	ad2ant2r 760	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ∈ On ∧ Lim 𝑥))
50		limsuc 7846	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
51	50	biimpa 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → suc 𝐴 ∈ 𝑥)
52		oveq2 7422	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = suc 𝐴 → (𝐶 ·o 𝑦) = (𝐶 ·o suc 𝐴))
53	52	ssiun2s 5007	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝐴 ∈ 𝑥 → (𝐶 ·o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
54	51, 53	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
55	54	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
56		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
57		omlim 8521	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
58	56, 57	mpanr1 716	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
59	58	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·o 𝑦))
60	55, 59	sseqtrrd 3968	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
61	49, 60	sylan 592	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o suc 𝐴) ⊆ (𝐶 ·o 𝑥))
62		omcl 8524	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
63		oaord1 8539	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·o 𝐴) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
64	62, 63	sylan 592	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
65	64	anabss1 679	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶)))
66	65	biimpa 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝐴) +o 𝐶))
67		omsuc 8514	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
68	67	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o suc 𝐴) = ((𝐶 ·o 𝐴) +o 𝐶))
69	66, 68	eleqtrrd 2863	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
70	69	adantrl 729	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
71	70	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝐴))
72	61, 71	sseldd 3932	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))
73	72	exp53 453	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐶 → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
74	73	com13 89	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))))
75	74	imp4c 429	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
76	75	a1dd 51	. . . . . . . . 9 ⊢ (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (∀𝑦 ∈ 𝑥 (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)))))
77	6, 10, 14, 18, 21, 47, 76	tfinds3 7862	. . . . . . . 8 ⊢ (𝐵 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
78	77	com23 87	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
79	78	exp4a 437	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
80	79	exp4a 437	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))))
81	2, 80	mpdd 44	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
82	81	com34 92	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ On → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
83	82	com24 96	. 2 ⊢ (𝐵 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
84	83	imp31 423	1 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  ∪ ciun 4951  Oncon0 6357  Lim wlim 6358  suc csuc 6359  (class class class)co 7414   +_o coa 8453   ·_o comu 8454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-oadd 8460  df-omul 8461

This theorem is used by:  omord2  8555  omcan  8557  odi  8567  omass  8568  oen0  8575  oeordi  8576  oeordsuc  8583  onexoegt  44086  omord2i  44143  oaomoencom  44159  cantnftermord  44162  omcl2  44175  omltoe  44248