Theorem nnmordi 6000

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 4255	. . . . . 6 ⊢ ((A ∈ B ∧ B ∈ 𝜔) → A ∈ 𝜔)
2	1	expcom 109	. . . . 5 ⊢ (B ∈ 𝜔 → (A ∈ B → A ∈ 𝜔))
3		eleq2 2083	. . . . . . . . . . 11 ⊢ (x = B → (A ∈ x ↔ A ∈ B))
4		oveq2 5444	. . . . . . . . . . . 12 ⊢ (x = B → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 B))
5	4	eleq2d 2089	. . . . . . . . . . 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 2083	. . . . . . . . . . 11 ⊢ (x = ∅ → (A ∈ x ↔ A ∈ ∅))
9		oveq2 5444	. . . . . . . . . . . 12 ⊢ (x = ∅ → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 ∅))
10	9	eleq2d 2089	. . . . . . . . . . 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 2083	. . . . . . . . . . 11 ⊢ (x = y → (A ∈ x ↔ A ∈ y))
13		oveq2 5444	. . . . . . . . . . . 12 ⊢ (x = y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 y))
14	13	eleq2d 2089	. . . . . . . . . . 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 2083	. . . . . . . . . . 11 ⊢ (x = suc y → (A ∈ x ↔ A ∈ suc y))
17		oveq2 5444	. . . . . . . . . . . 12 ⊢ (x = suc y → (𝐶 ·𝑜 x) = (𝐶 ·𝑜 suc y))
18	17	eleq2d 2089	. . . . . . . . . . 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 3205	. . . . . . . . . . . 12 ⊢ ¬ A ∈ ∅
21	20	pm2.21i 562	. . . . . . . . . . 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 4089	. . . . . . . . . . . . . . . 16 ⊢ (A ∈ suc y → (A ∈ y ∨ A = y))
24		nnmcl 5975	. . . . . . . . . . . . . . . . . 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 5997	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (𝐶 ·𝑜 y) ⊆ ((𝐶 ·𝑜 y) +𝑜 𝐶))
28	27	sseld 2921	. . . . . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . . 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 5968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 ·𝑜 y) ∈ 𝜔 → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
33	32	ad2antrr 460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 y) +𝑜 ∅) = (𝐶 ·𝑜 y))
34		nnaordi 5992	. . . . . . . . . . . . . . . . . . . . . . 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 2097	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶 ·𝑜 y) ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 y) ∈ ((𝐶 ·𝑜 y) +𝑜 𝐶))
38		oveq2 5444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (A = y → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 y))
39	38	eleq1d 2088	. . . . . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . . . . 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 624	. . . . . . . . . . . . . . . . 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 5971	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ 𝜔 ∧ y ∈ 𝜔) → (𝐶 ·𝑜 suc y) = ((𝐶 ·𝑜 y) +𝑜 𝐶))
46	45	eleq2d 2089	. . . . . . . . . . . . . . . 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 4251	. . . . . . . . 9 ⊢ (x ∈ 𝜔 → (((A ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ x → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 x))))
54	7, 53	vtoclga 2596	. . . . . . . 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 616   = wceq 1228   ∈ wcel 1374  ∅c0 3201  suc csuc 4051  𝜔com 4240  (class class class)co 5436   +_𝑜 coa 5913   ·_𝑜 comu 5914

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921

This theorem is referenced by:  nnmord  6001  nnm00  6013  mulclpi  6188