Theorem nnmord 6467

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 6466	. . . . . 6 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
2	1	ex 114	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
3	2	com23 78	. . . 4 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
4	3	impd 252	. . 3 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
5	4	3adant1 1000	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
6		ne0i 3401	. . . . . . . 8 ⊢ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (𝐶 ·o 𝐵) ≠ ∅)
7		nnm0r 6429	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
8		oveq1 5834	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
9	8	eqeq1d 2166	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((𝐶 ·o 𝐵) = ∅ ↔ (∅ ·o 𝐵) = ∅))
10	7, 9	syl5ibrcom 156	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐶 = ∅ → (𝐶 ·o 𝐵) = ∅))
11	10	necon3d 2371	. . . . . . . 8 ⊢ (𝐵 ∈ ω → ((𝐶 ·o 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
12	6, 11	syl5 32	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅))
13	12	adantr 274	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅))
14		nn0eln0 4582	. . . . . . 7 ⊢ (𝐶 ∈ ω → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15	14	adantl 275	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
16	13, 15	sylibrd 168	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶))
17	16	3adant1 1000	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶))
18		oveq2 5835	. . . . . . . . . 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 6466	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
21	20	3adantl2 1139	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
22	19, 21	orim12d 776	. . . . . . . 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 621	. . . . . . 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 987	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
25		simpl1 985	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
26		nnmcl 6431	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·o 𝐴) ∈ ω)
27	24, 25, 26	syl2anc 409	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ω)
28		simpl2 986	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
29		nnmcl 6431	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o 𝐵) ∈ ω)
30	24, 28, 29	syl2anc 409	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐵) ∈ ω)
31		nntri2 6444	. . . . . . . 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 409	. . . . . . 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 6444	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
34	25, 28, 33	syl2anc 409	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
35	23, 32, 34	3imtr4d 202	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵))
36	35	ex 114	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵)))
37	36	com23 78	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (∅ ∈ 𝐶 → 𝐴 ∈ 𝐵)))
38	17, 37	mpdd 41	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵))
39	38, 17	jcad 305	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
40	5, 39	impbid 128	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963   = wceq 1335   ∈ wcel 2128   ≠ wne 2327  ∅c0 3395  ωcom 4552  (class class class)co 5827   ·_o comu 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-iord 4329  df-on 4331  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-irdg 6320  df-oadd 6370  df-omul 6371

This theorem is referenced by:  nnmword  6468  ltmpig  7262