Theorem nnmordi 8613

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 7869	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
2	1	expcom 418	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω))
3		eleq2 2852	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
4		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝐵))
5	4	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
6	3, 5	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
7	6	imbi2d 343	. . . . . . . . 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 2852	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ∅))
9		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝐶 ·o 𝑥) = (𝐶 ·o ∅))
10	9	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
11	8, 10	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))))
12		eleq2 2852	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
13		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o 𝑦))
14	13	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦)))
15	12, 14	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦))))
16		eleq2 2852	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝑦))
17		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (𝐶 ·o 𝑥) = (𝐶 ·o suc 𝑦))
18	17	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦)))
19	16, 18	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥)) ↔ (𝐴 ∈ suc 𝑦 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦))))
20		noel 4291	. . . . . . . . . . . 12 ⊢ ¬ 𝐴 ∈ ∅
21	20	pm2.21i 120	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅))
22	21	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ ∅ → (𝐶 ·o 𝐴) ∈ (𝐶 ·o ∅)))
23		elsuci 6430	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
24		nnmcl 8594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·o 𝑦) ∈ ω)
25		simpl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → 𝐶 ∈ ω)
26	24, 25	jca 520	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω))
27		nnaword1 8611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝑦) ⊆ ((𝐶 ·o 𝑦) +o 𝐶))
28	27	sseld 3936	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑦) → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
29	28	imim2d 58	. . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 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 8586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 ·o 𝑦) ∈ ω → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
33	32	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝑦) +o ∅) = (𝐶 ·o 𝑦))
34		nnaordi 8600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐶 ∈ ω ∧ (𝐶 ·o 𝑦) ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
35	34	ancoms 463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
36	35	imp 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝑦) +o ∅) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
37	33, 36	eqeltrrd 2864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶))
38		oveq2 7418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝑦 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝑦))
39	38	eleq1d 2848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 = 𝑦 → ((𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶) ↔ (𝐶 ·o 𝑦) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
40	37, 39	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ·o 𝑦) ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝑦 → (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
41	40	adantrr 729	. . . . . . . . . . . . . . . . . 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 872	. . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . 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 35	. . . . . . . . . . . . . . 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 8589	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 ·o suc 𝑦) = ((𝐶 ·o 𝑦) +o 𝐶))
46	45	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o suc 𝑦) ↔ (𝐶 ·o 𝐴) ∈ ((𝐶 ·o 𝑦) +o 𝐶)))
47	46	adantr 485	. . . . . . . . . . . . . . 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 262	. . . . . . . . . . . . . 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 441	. . . . . . . . . . . . 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 33	. . . . . . . . . . . 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 495	. . . . . . . . . . 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 415	. . . . . . . . . 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 7891	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝑥 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝑥))))
54	7, 53	vtoclga 3541	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
55	54	com23 87	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
56	55	exp4a 436	. . . . . 6 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
57	56	exp4a 436	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (𝐴 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))))
58	2, 57	mpdd 44	. . . 4 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
59	58	com34 92	. . 3 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → (∅ ∈ 𝐶 → (𝐶 ∈ ω → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
60	59	com24 96	. 2 ⊢ (𝐵 ∈ ω → (𝐶 ∈ ω → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))))
61	60	imp31 422	1 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∅c0 4286  suc csuc 6362  (class class class)co 7410  ωcom 7858   +_o coa 8446   ·_o comu 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-oadd 8453  df-omul 8454

This theorem is referenced by:  nnmord  8614  mulclpi  10873