Theorem nnmordi 6779

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 4748	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
2	1	expcom 116	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
3		eleq2 2302	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
4		oveq2 6083	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
5	4	eleq2d 2308	. . . . . . . . . . 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 2302	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
9		oveq2 6083	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
10	9	eleq2d 2308	. . . . . . . . . . 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 2302	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
13		oveq2 6083	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
14	13	eleq2d 2308	. . . . . . . . . . 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 2302	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
17		oveq2 6083	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
18	17	eleq2d 2308	. . . . . . . . . . 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 3525	. . . . . . . . . . . 12 ⊢ ¬ 𝐴 ∈ ∅
21	20	pm2.21i 655	. . . . . . . . . . 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 4543	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
24		nnmcl 6744	. . . . . . . . . . . . . . . . . 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 6776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
28	27	sseld 3247	. . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 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 6737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 ·o 𝑦) ∈ ω → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
33	32	ad2antrr 492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
34		nnaordi 6771	. . . . . . . . . . . . . . . . . . . . . . 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 2316	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
38		oveq2 6083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
39	38	eleq1d 2307	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . 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 6740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
46	45	eleq2d 2308	. . . . . . . . . . . . . . . 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 4743	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
54	7, 53	vtoclga 2889	. . . . . . . 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 720   = wceq 1402   ∈ wcel 2209  ∅c0 3520  suc csuc 4505  ωcom 4732  (class class class)co 6075   +_o coa 6674   ·_o comu 6675

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682

This theorem is referenced by:  nnmord  6780  nnm00  6793  mulclpi  7685