Theorem omordi 7646

Description: Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (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 5748	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	ex 450	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3		eleq2 2690	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
4		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 ∅))
5	4	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅)))
6	3, 5	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 ∅))))
7		eleq2 2690	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
8		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝑦))
9	8	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦)))
10	7, 9	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦))))
11		eleq2 2690	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
12		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 suc 𝑦))
13	12	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦)))
14	11, 13	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦))))
15		eleq2 2690	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
16		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐶 ·𝑜 𝑥) = (𝐶 ·𝑜 𝐵))
17	16	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
18	15, 17	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥)) ↔ (𝐴 ∈ 𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
19		noel 3919	. . . . . . . . . . 11 ⊢ ¬ 𝐴 ∈ ∅
20	19	pm2.21i 116	. . . . . . . . . 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 5791	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
23		omcl 7616	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·𝑜 𝑦) ∈ On)
24		simpl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → 𝐶 ∈ On)
25	23, 24	jca 554	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On))
26		oaword1 7632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) → (𝐶 ·𝑜 𝑦) ⊆ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
27	26	sseld 3602	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑦) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
28	27	imim2d 57	. . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . . 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 752	. . . . . . . . . . . . . . . . 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 7631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
32	31	biimpa 501	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
33		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝑦))
34	33	eleq1d 2686	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑦 → ((𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶) ↔ (𝐶 ·𝑜 𝑦) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
35	32, 34	syl5ibrcom 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ·𝑜 𝑦) ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
36	35	adantrr 753	. . . . . . . . . . . . . . . . 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 395	. . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 34	. . . . . . . . . . . . . 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 7606	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 ·𝑜 suc 𝑦) = ((𝐶 ·𝑜 𝑦) +𝑜 𝐶))
41	40	eleq2d 2687	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝑦) ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝑦) +𝑜 𝐶)))
42	41	adantr 481	. . . . . . . . . . . . . 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 249	. . . . . . . . . . . . 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 640	. . . . . . . . . . . 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 32	. . . . . . . . . . 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 483	. . . . . . . . . 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 447	. . . . . . . . 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 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ∈ On ∧ Lim 𝑥))
49	48	ad2ant2r 783	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ∈ On ∧ Lim 𝑥))
50		limsuc 7049	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥))
51	50	biimpa 501	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → suc 𝐴 ∈ 𝑥)
52		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = suc 𝐴 → (𝐶 ·𝑜 𝑦) = (𝐶 ·𝑜 suc 𝐴))
53	52	ssiun2s 4564	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝐴 ∈ 𝑥 → (𝐶 ·𝑜 suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·𝑜 𝑦))
54	51, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ 𝑥) → (𝐶 ·𝑜 suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·𝑜 𝑦))
55	54	adantll 750	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·𝑜 suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 ·𝑜 𝑦))
56		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
57		omlim 7613	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·𝑜 𝑦))
58	56, 57	mpanr1 719	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·𝑜 𝑦))
59	58	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 ·𝑜 𝑦))
60	55, 59	sseqtr4d 3642	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ Lim 𝑥) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝑥))
61	49, 60	sylan 488	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝑥))
62		omcl 7616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·𝑜 𝐴) ∈ On)
63		oaord1 7631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·𝑜 𝐴) ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶)))
64	62, 63	sylan 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶)))
65	64	anabss1 855	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐶 ↔ (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶)))
66	65	biimpa 501	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ ((𝐶 ·𝑜 𝐴) +𝑜 𝐶))
67		omsuc 7606	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·𝑜 suc 𝐴) = ((𝐶 ·𝑜 𝐴) +𝑜 𝐶))
68	67	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 suc 𝐴) = ((𝐶 ·𝑜 𝐴) +𝑜 𝐶))
69	66, 68	eleqtrrd 2704	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝐴))
70	69	adantrl 752	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝐴))
71	70	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 suc 𝐴))
72	61, 71	sseldd 3604	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐶)) ∧ 𝐴 ∈ 𝑥) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))
73	72	exp53 647	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → (𝐴 ∈ On → (Lim 𝑥 → (∅ ∈ 𝐶 → (𝐴 ∈ 𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))))))
74	73	com13 88	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴 ∈ 𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))))))
75	74	imp4c 617	. . . . . . . . . 10 ⊢ (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝑥 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝑥))))
76	75	a1dd 50	. . . . . . . . 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 7064	. . . . . . . 8 ⊢ (𝐵 ∈ On → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
78	77	com23 86	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
79	78	exp4a 633	. . . . . 6 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
80	79	exp4a 633	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))))
81	2, 80	mpdd 43	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
82	81	com34 91	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ On → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
83	82	com24 95	. 2 ⊢ (𝐵 ∈ On → (𝐶 ∈ On → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))))
84	83	imp31 448	1 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650   +_𝑜 coa 7557   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omord2  7647  omcan  7649  odi  7659  omass  7660  oen0  7666  oeordi  7667  oeordsuc  7674