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Theorem ordpinq 10100

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 5428	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵))
2		df-ltnq 10075	. . . 4 ⊢ <_Q = ( <_pQ ∩ (Q × Q))
3	2	breqi 4892	. . 3 ⊢ (𝐴 <_Q 𝐵 ↔ 𝐴( <_pQ ∩ (Q × Q))𝐵)
4	1, 3	syl6bbr 281	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ 𝐴 <_Q 𝐵))
5		relxp 5373	. . . . 5 ⊢ Rel (N × N)
6		elpqn 10082	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
7		1st2nd 7493	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
8	5, 6, 7	sylancr 581	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
9		elpqn 10082	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
10		1st2nd 7493	. . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
11	5, 9, 10	sylancr 581	. . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
12	8, 11	breqan12d 4902	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_pQ 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ <_pQ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
13		ordpipq 10099	. . 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 279	. 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 273	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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ⟨cop 4404 class class class wbr 4886 × cxp 5353 Rel wrel 5360 ‘cfv 6135 (class class class)co 6922 1^st c1st 7443 2^nd c2nd 7444 Ncnpi 10001 ·_N cmi 10003 <_N clti 10004 <_pQ cltpq 10007 Qcnq 10009 <_Q cltq 10015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-omul 7848 df-ni 10029 df-mi 10031 df-lti 10032 df-ltpq 10067 df-nq 10069 df-ltnq 10075

This theorem is referenced by: ltsonq 10126 lterpq 10127 ltanq 10128 ltmnq 10129 ltexnq 10132 archnq 10137

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