Theorem nnmordi 6025

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 4271	. . . . . 6 ⊢ ((A ∈ B ∧ B ∈ 𝜔) → A ∈ 𝜔)
2	1	expcom 109	. . . . 5 ⊢ (B ∈ 𝜔 → (A ∈ B → A ∈ 𝜔))
3		eleq2 2098	. . . . . . . . . . 11 ⊢ (x = B → (A ∈ x ↔ A ∈ B))
4		oveq2 5463	. . . . . . . . . . . 12 ⊢ (x = B → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 B))
5	4	eleq2d 2104	. . . . . . . . . . 11 ⊢ (x = B → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))
6	3, 5	imbi12d 223	. . . . . . . . . 10 ⊢ (x = B → ((A ∈ x → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x)) ↔ (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))
7	6	imbi2d 219	. . . . . . . . 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 2098	. . . . . . . . . . 11 ⊢ (x = ∅ → (A ∈ x ↔ A ∈ ∅))
9		oveq2 5463	. . . . . . . . . . . 12 ⊢ (x = ∅ → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 ∅))
10	9	eleq2d 2104	. . . . . . . . . . 11 ⊢ (x = ∅ → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 ∅)))
11	8, 10	imbi12d 223	. . . . . . . . . 10 ⊢ (x = ∅ → ((A ∈ x → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x)) ↔ (A ∈ ∅ → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 ∅))))
12		eleq2 2098	. . . . . . . . . . 11 ⊢ (x = y → (A ∈ x ↔ A ∈ y))
13		oveq2 5463	. . . . . . . . . . . 12 ⊢ (x = y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 y))
14	13	eleq2d 2104	. . . . . . . . . . 11 ⊢ (x = y → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 y)))
15	12, 14	imbi12d 223	. . . . . . . . . 10 ⊢ (x = y → ((A ∈ x → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x)) ↔ (A ∈ y → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 y))))
16		eleq2 2098	. . . . . . . . . . 11 ⊢ (x = suc y → (A ∈ x ↔ A ∈ suc y))
17		oveq2 5463	. . . . . . . . . . . 12 ⊢ (x = suc y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 suc y))
18	17	eleq2d 2104	. . . . . . . . . . 11 ⊢ (x = suc y → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x) ↔ (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 suc y)))
19	16, 18	imbi12d 223	. . . . . . . . . 10 ⊢ (x = suc y → ((A ∈ x → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x)) ↔ (A ∈ suc y → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 suc y))))
20		noel 3222	. . . . . . . . . . . 12 ⊢ ¬ A ∈ ∅
21	20	pm2.21i 574	. . . . . . . . . . 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 4106	. . . . . . . . . . . . . . . 16 ⊢ (A ∈ suc y → (A ∈ y ∨ A = y))
24		nnmcl 5999	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ 𝜔 ∧ y ∈ 𝜔) → (𝐶 ·𝑜 y) ∈ 𝜔)
25		simpl 102	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ 𝜔 ∧ y ∈ 𝜔) → 𝐶 ∈ 𝜔)
26	24, 25	jca 290	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ 𝜔 ∧ y ∈ 𝜔) → ((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔))
27		nnaword1 6022	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (𝐶 ·𝑜 y) ⊆ ((𝐶 ·𝑜 y) +𝑜 𝐶))
28	27	sseld 2938	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 y) → (𝐶 ·𝑜 A) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶)))
29	28	imim2d 48	. . . . . . . . . . . . . . . . . . . 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 115	. . . . . . . . . . . . . . . . . . 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 447	. . . . . . . . . . . . . . . . . 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 5992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 ·𝑜 y) ∈ 𝜔 → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
33	32	ad2antrr 457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
34		nnaordi 6017	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐶 ∈ 𝜔 ∧ (𝐶 ·𝑜 y) ∈ 𝜔) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 y) +𝑜 ∅) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶)))
35	34	ancoms 255	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 y) +𝑜 ∅) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶)))
36	35	imp 115	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 y) +𝑜 ∅) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶))
37	33, 36	eqeltrrd 2112	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 y) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶))
38		oveq2 5463	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (A = y → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 y))
39	38	eleq1d 2103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (A = y → ((𝐶 ·𝑜 A) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶) ↔ (𝐶 ·𝑜 y) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶)))
40	37, 39	syl5ibrcom 146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A = y → (𝐶 ·𝑜 A) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶)))
41	40	adantrr 448	. . . . . . . . . . . . . . . . . 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 636	. . . . . . . . . . . . . . . . 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 267	. . . . . . . . . . . . . . . 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 28	. . . . . . . . . . . . . . 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 5995	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ 𝜔 ∧ y ∈ 𝜔) → (𝐶 ·𝑜 suc y) = ((𝐶 ·𝑜 y) +𝑜 𝐶))
46	45	eleq2d 2104	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ 𝜔 ∧ y ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 suc y) ↔ (𝐶 ·𝑜 A) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶)))
47	46	adantr 261	. . . . . . . . . . . . . . 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 158	. . . . . . . . . . . . . 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 354	. . . . . . . . . . . . 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 27	. . . . . . . . . . . 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 263	. . . . . . . . . . 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 242	. . . . . . . . . 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 4267	. . . . . . . . 9 ⊢ (x ∈ 𝜔 → (((A ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ x → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x))))
54	7, 53	vtoclga 2613	. . . . . . . 8 ⊢ (B ∈ 𝜔 → (((A ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))
55	54	com23 72	. . . . . . 7 ⊢ (B ∈ 𝜔 → (A ∈ B → (((A ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))
56	55	exp4a 348	. . . . . 6 ⊢ (B ∈ 𝜔 → (A ∈ B → ((A ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))))
57	56	exp4a 348	. . . . 5 ⊢ (B ∈ 𝜔 → (A ∈ B → (A ∈ 𝜔 → (𝐶 ∈ 𝜔 → (∅ ∈ 𝐶 → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))))
58	2, 57	mpdd 36	. . . 4 ⊢ (B ∈ 𝜔 → (A ∈ B → (𝐶 ∈ 𝜔 → (∅ ∈ 𝐶 → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))))
59	58	com34 77	. . 3 ⊢ (B ∈ 𝜔 → (A ∈ B → (∅ ∈ 𝐶 → (𝐶 ∈ 𝜔 → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))))
60	59	com24 81	. 2 ⊢ (B ∈ 𝜔 → (𝐶 ∈ 𝜔 → (∅ ∈ 𝐶 → (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))))
61	60	imp31 243	1 ⊢ (((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  ∅c0 3218  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +_𝑜 coa 5937   ·_𝑜 comu 5938

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945

This theorem is referenced by:  nnmord  6026  nnm00  6038  mulclpi  6312