Theorem ordpinq 10924

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 5738	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵))
2		df-ltnq 10899	. . . 4 ⊢ <Q = ( <pQ ∩ (Q × Q))
3	2	breqi 5116	. . 3 ⊢ (𝐴 <Q 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵)
4	1, 3	bitr4di 292	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴 <Q 𝐵))
5		relxp 5677	. . . . 5 ⊢ Rel (N × N)
6		elpqn 10906	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 8032	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
8	5, 6, 7	sylancr 598	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
9		elpqn 10906	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 8032	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
11	5, 9, 10	sylancr 598	. . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
12	8, 11	breqan12d 5126	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩))
13		ordpipq 10923	. . 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 290	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
15	4, 14	bitr3d 284	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∩ cin 3912  ⟨cop 4597   class class class wbr 5110   × cxp 5657  Rel wrel 5664  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  Ncnpi 10825   ·_N cmi 10827   <_N clti 10828   <_pQ cltpq 10831  Qcnq 10833   <_Q cltq 10839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-omul 8454  df-ni 10853  df-mi 10855  df-lti 10856  df-ltpq 10891  df-nq 10893  df-ltnq 10899

This theorem is referenced by:  ltsonq  10950  lterpq  10951  ltanq  10952  ltmnq  10953  ltexnq  10956  archnq  10961