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Theorem ordpinq 10837

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 ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) <_N ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))))

**Proof of Theorem ordpinq**
Step	Hyp	Ref	Expression
1		brinxp 5698	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵))
2		df-ltnq 10812	. . . 4 ⊢ <_Q = ( <_pQ ∩ (Q × Q))
3	2	breqi 5098	. . 3 ⊢ (𝐴 <_Q 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵)
4	1, 3	bitr4di 289	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴 <_Q 𝐵))
5		relxp 5637	. . . . 5 ⊢ Rel (N × N)
6		elpqn 10819	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 7974	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
8	5, 6, 7	sylancr 587	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
9		elpqn 10819	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 7974	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
11	5, 9, 10	sylancr 587	. . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
12	8, 11	breqan12d 5108	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ <_pQ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
13		ordpipq 10836	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ <_pQ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) <_N ((1^st ‘𝐵) ·_N (2^nd ‘𝐴)))
14	12, 13	bitrdi 287	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) <_N ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))))
15	4, 14	bitr3d 281	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_Q 𝐵 ↔ ((1^st ‘𝐴) ·_N (2^nd ‘𝐵)) <_N ((1^st ‘𝐵) ·_N (2^nd ‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∩ cin 3902 ⟨cop 4583 class class class wbr 5092 × cxp 5617 Rel wrel 5624 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 Ncnpi 10738 ·_N cmi 10740 <_N clti 10741 <_pQ cltpq 10744 Qcnq 10746 <_Q cltq 10752

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-omul 8393 df-ni 10766 df-mi 10768 df-lti 10769 df-ltpq 10804 df-nq 10806 df-ltnq 10812

This theorem is referenced by: ltsonq 10863 lterpq 10864 ltanq 10865 ltmnq 10866 ltexnq 10869 archnq 10874

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