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Theorem ordpinq 9803

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 5215	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵))
2		df-ltnq 9778	. . . 4 ⊢ <_Q = ( <_pQ ∩ (Q × Q))
3	2	breqi 4691	. . 3 ⊢ (𝐴 <_Q 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵)
4	1, 3	syl6bbr 278	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴 <_Q 𝐵))
5		relxp 5160	. . . . 5 ⊢ Rel (N × N)
6		elpqn 9785	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 7258	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
8	5, 6, 7	sylancr 696	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
9		elpqn 9785	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 7258	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
11	5, 9, 10	sylancr 696	. . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
12	8, 11	breqan12d 4701	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ <_pQ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
13		ordpipq 9802	. . 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	syl6bb 276	. 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 270	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 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∩ cin 3606 ⟨cop 4216 class class class wbr 4685 × cxp 5141 Rel wrel 5148 ‘cfv 5926 (class class class)co 6690 1^st c1st 7208 2^nd c2nd 7209 Ncnpi 9704 ·_N cmi 9706 <_N clti 9707 <_pQ cltpq 9710 Qcnq 9712 <_Q cltq 9718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-omul 7610 df-ni 9732 df-mi 9734 df-lti 9735 df-ltpq 9770 df-nq 9772 df-ltnq 9778

This theorem is referenced by: ltsonq 9829 lterpq 9830 ltanq 9831 ltmnq 9832 ltexnq 9835 archnq 9840

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