Theorem nnmordi 6660

Description: Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnmordi	⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Proof of Theorem nnmordi

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

Step	Hyp	Ref	Expression
1		elnn 4697	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
2	1	expcom 116	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
3		eleq2 2293	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
4		oveq2 6008	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
5	4	eleq2d 2299	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
6	3, 5	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
7	6	imbi2d 230	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))) ↔ (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
8		eleq2 2293	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
9		oveq2 6008	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
10	9	eleq2d 2299	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
11	8, 10	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
12		eleq2 2293	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
13		oveq2 6008	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
14	13	eleq2d 2299	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
15	12, 14	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
16		eleq2 2293	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
17		oveq2 6008	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
18	17	eleq2d 2299	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
19	16, 18	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
20		noel 3495	. . . . . . . . . . . 12 ⊢ ¬ 𝐴 ∈ ∅
21	20	pm2.21i 649	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))
22	21	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
23		elsuci 4493	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
24		nnmcl 6625	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·o 𝑦) ∈ ω)
25		simpl 109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → 𝐶 ∈ ω)
26	24, 25	jca 306	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω))
27		nnaword1 6657	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
28	27	sseld 3223	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
29	28	imim2d 54	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶))))
30	29	imp 124	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
31	30	adantrl 478	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
32		nna0 6618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 ·o 𝑦) ∈ ω → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
33	32	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
34		nnaordi 6652	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐶 ∈ ω ∧ (𝐶 ·o 𝑦) ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
35	34	ancoms 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
36	35	imp 124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
37	33, 36	eqeltrrd 2307	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
38		oveq2 6008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
39	38	eleq1d 2298	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
40	37, 39	syl5ibrcom 157	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
41	40	adantrr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
42	31, 41	jaod 722	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
43	26, 42	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
44	23, 43	syl5 32	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
45		nnmsuc 6621	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
46	45	eleq2d 2299	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
47	46	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
48	44, 47	sylibrd 169	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ ω ∧ 𝑦 ∈ ω) ∧ (∅ ∈ 𝐶 ∧ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
49	48	exp43 372	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ω → (𝑦 ∈ ω → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
50	49	com12 30	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
51	50	adantld 278	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))))
52	51	impd 254	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)) → (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))))
53	11, 15, 19, 22, 52	finds2 4692	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
54	7, 53	vtoclga 2867	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
55	54	com23 78	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
56	55	exp4a 366	. . . . . 6 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
57	56	exp4a 366	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (𝐴 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))))
58	2, 57	mpdd 41	. . . 4 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
59	58	com34 83	. . 3 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ ω → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
60	59	com24 87	. 2 ⊢ (𝐵 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
61	60	imp31 256	1 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∅c0 3491  suc csuc 4455  ωcom 4681  (class class class)co 6000   +_o coa 6557   ·_o comu 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-oadd 6564  df-omul 6565

This theorem is referenced by:  nnmord  6661  nnm00  6674  mulclpi  7511