Theorem ordpinq 10940

Description: Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ordpinq	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))

Proof of Theorem ordpinq

Step	Hyp	Ref	Expression
1		brinxp 5753	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵))
2		df-ltnq 10915	. . . 4 ⊢ <Q = ( <pQ ∩ (Q × Q))
3	2	breqi 5153	. . 3 ⊢ (𝐴 <Q 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵)
4	1, 3	bitr4di 288	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴 <Q 𝐵))
5		relxp 5693	. . . . 5 ⊢ Rel (N × N)
6		elpqn 10922	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 8027	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
8	5, 6, 7	sylancr 585	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
9		elpqn 10922	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 8027	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
11	5, 9, 10	sylancr 585	. . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
12	8, 11	breqan12d 5163	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩))
13		ordpipq 10939	. . 3 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
14	12, 13	bitrdi 286	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
15	4, 14	bitr3d 280	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ∩ cin 3946  ⟨cop 4633   class class class wbr 5147   × cxp 5673  Rel wrel 5680  ‘cfv 6542  (class class class)co 7411  1^st c1st 7975  2^nd c2nd 7976  Ncnpi 10841   ·_N cmi 10843   <_N clti 10844   <_pQ cltpq 10847  Qcnq 10849   <_Q cltq 10855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-omul 8473  df-ni 10869  df-mi 10871  df-lti 10872  df-ltpq 10907  df-nq 10909  df-ltnq 10915

This theorem is referenced by:  ltsonq  10966  lterpq  10967  ltanq  10968  ltmnq  10969  ltexnq  10972  archnq  10977