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Theorem ordpinq 10699

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 5665	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵))
2		df-ltnq 10674	. . . 4 ⊢ <_Q = ( <_pQ ∩ (Q × Q))
3	2	breqi 5080	. . 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 5607	. . . . 5 ⊢ Rel (N × N)
6		elpqn 10681	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 7880	. . . . 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 10681	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 7880	. . . . 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 5090	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ <_pQ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
13		ordpipq 10698	. . 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 280	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⟨cop 4567 class class class wbr 5074 × cxp 5587 Rel wrel 5594 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Ncnpi 10600 ·_N cmi 10602 <_N clti 10603 <_pQ cltpq 10606 Qcnq 10608 <_Q cltq 10614

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-omul 8302 df-ni 10628 df-mi 10630 df-lti 10631 df-ltpq 10666 df-nq 10668 df-ltnq 10674

This theorem is referenced by: ltsonq 10725 lterpq 10726 ltanq 10727 ltmnq 10728 ltexnq 10731 archnq 10736

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