Theorem ordpinq 10928

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 5741	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵))
2		df-ltnq 10903	. . . 4 ⊢ <Q = ( <pQ ∩ (Q × Q))
3	2	breqi 5117	. . 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 5680	. . . . 5 ⊢ Rel (N × N)
6		elpqn 10910	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 8036	. . . . 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 10910	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 8036	. . . . 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 5127	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩))
13		ordpipq 10927	. . 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 3910  ⟨cop 4598   class class class wbr 5111   × cxp 5660  Rel wrel 5667  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  Ncnpi 10829   ·_N cmi 10831   <_N clti 10832   <_pQ cltpq 10835  Qcnq 10837   <_Q cltq 10843

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 5259  ax-nul 5271  ax-pr 5405  ax-un 7733

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 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-omul 8458  df-ni 10857  df-mi 10859  df-lti 10860  df-ltpq 10895  df-nq 10897  df-ltnq 10903

This theorem is referenced by:  ltsonq  10954  lterpq  10955  ltanq  10956  ltmnq  10957  ltexnq  10960  archnq  10965