Theorem nnmord 6001

Description: Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

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

Proof of Theorem nnmord

Step	Hyp	Ref	Expression
1		nnmordi 6000	. . . . . 6 ⊢ (((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))
2	1	ex 108	. . . . 5 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 → (A ∈ B → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))
3	2	com23 72	. . . 4 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (A ∈ B → (∅ ∈ 𝐶 → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B))))
4	3	impd 242	. . 3 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((A ∈ B ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))
5	4	3adant1 910	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((A ∈ B ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))
6		ne0i 3207	. . . . . . . 8 ⊢ ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → (𝐶 ·𝑜 B) ≠ ∅)
7		nnm0r 5973	. . . . . . . . . 10 ⊢ (B ∈ 𝜔 → (∅ ·𝑜 B) = ∅)
8		oveq1 5443	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 B) = (∅ ·𝑜 B))
9	8	eqeq1d 2030	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((𝐶 ·𝑜 B) = ∅ ↔ (∅ ·𝑜 B) = ∅))
10	7, 9	syl5ibrcom 146	. . . . . . . . 9 ⊢ (B ∈ 𝜔 → (𝐶 = ∅ → (𝐶 ·𝑜 B) = ∅))
11	10	necon3d 2227	. . . . . . . 8 ⊢ (B ∈ 𝜔 → ((𝐶 ·𝑜 B) ≠ ∅ → 𝐶 ≠ ∅))
12	6, 11	syl5 28	. . . . . . 7 ⊢ (B ∈ 𝜔 → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → 𝐶 ≠ ∅))
13	12	adantr 261	. . . . . 6 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → 𝐶 ≠ ∅))
14		nn0eln0 4268	. . . . . . 7 ⊢ (𝐶 ∈ 𝜔 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15	14	adantl 262	. . . . . 6 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
16	13, 15	sylibrd 158	. . . . 5 ⊢ ((B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → ∅ ∈ 𝐶))
17	16	3adant1 910	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → ∅ ∈ 𝐶))
18		oveq2 5444	. . . . . . . . . 10 ⊢ (A = B → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 B))
19	18	a1i 9	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A = B → (𝐶 ·𝑜 A) = (𝐶 ·𝑜 B)))
20		nnmordi 6000	. . . . . . . . . 10 ⊢ (((A ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (B ∈ A → (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A)))
21	20	3adantl2 1049	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (B ∈ A → (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A)))
22	19, 21	orim12d 687	. . . . . . . 8 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((A = B ∨ B ∈ A) → ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) ∨ (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A))))
23	22	con3d 548	. . . . . . 7 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) ∨ (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A)) → ¬ (A = B ∨ B ∈ A)))
24		simpl3 897	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ 𝜔)
25		simpl1 895	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → A ∈ 𝜔)
26		nnmcl 5975	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝜔 ∧ A ∈ 𝜔) → (𝐶 ·𝑜 A) ∈ 𝜔)
27	24, 25, 26	syl2anc 393	. . . . . . . 8 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 A) ∈ 𝜔)
28		simpl2 896	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → B ∈ 𝜔)
29		nnmcl 5975	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝜔 ∧ B ∈ 𝜔) → (𝐶 ·𝑜 B) ∈ 𝜔)
30	24, 28, 29	syl2anc 393	. . . . . . . 8 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 B) ∈ 𝜔)
31		nntri2 5988	. . . . . . . 8 ⊢ (((𝐶 ·𝑜 A) ∈ 𝜔 ∧ (𝐶 ·𝑜 B) ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) ↔ ¬ ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) ∨ (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A))))
32	27, 30, 31	syl2anc 393	. . . . . . 7 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) ↔ ¬ ((𝐶 ·𝑜 A) = (𝐶 ·𝑜 B) ∨ (𝐶 ·𝑜 B) ∈ (𝐶 ·𝑜 A))))
33		nntri2 5988	. . . . . . . 8 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
34	25, 28, 33	syl2anc 393	. . . . . . 7 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
35	23, 32, 34	3imtr4d 192	. . . . . 6 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → A ∈ B))
36	35	ex 108	. . . . 5 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → A ∈ B)))
37	36	com23 72	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → (∅ ∈ 𝐶 → A ∈ B)))
38	17, 37	mpdd 36	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → A ∈ B))
39	38, 17	jcad 291	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B) → (A ∈ B ∧ ∅ ∈ 𝐶)))
40	5, 39	impbid 120	1 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → ((A ∈ B ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 A) ∈ (𝐶 ·𝑜 B)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   ∧ w3a 873   = wceq 1228   ∈ wcel 1374   ≠ wne 2186  ∅c0 3201  𝜔com 4240  (class class class)co 5436   ·_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-3or 874  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:  nnmword  6002  ltmpig  6199