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Theorem ordpinq 10861

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 5700	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵))
2		df-ltnq 10836	. . . 4 ⊢ <_Q = ( <_pQ ∩ (Q × Q))
3	2	breqi 5081	. . 3 ⊢ (𝐴 <_Q 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵)
4	1, 3	bitr4di 291	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴 <_Q 𝐵))
5		relxp 5639	. . . . 5 ⊢ Rel (N × N)
6		elpqn 10843	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 7985	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
8	5, 6, 7	sylancr 594	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
9		elpqn 10843	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 7985	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
11	5, 9, 10	sylancr 594	. . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
12	8, 11	breqan12d 5091	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ <_pQ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
13		ordpipq 10860	. . 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 289	. 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 283	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 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∩ cin 3884 ⟨cop 4564 class class class wbr 5075 × cxp 5619 Rel wrel 5626 ‘cfv 6489 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Ncnpi 10762 ·_N cmi 10764 <_N clti 10765 <_pQ cltpq 10768 Qcnq 10770 <_Q cltq 10776

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-fv 6497 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-omul 8404 df-ni 10790 df-mi 10792 df-lti 10793 df-ltpq 10828 df-nq 10830 df-ltnq 10836

This theorem is referenced by: ltsonq 10887 lterpq 10888 ltanq 10889 ltmnq 10890 ltexnq 10893 archnq 10898

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