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Mirrors > Home > MPE Home > Th. List > ordpinq | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ordpinq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brinxp 5756 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵)) | |
2 | df-ltnq 10942 | . . . 4 ⊢ <Q = ( <pQ ∩ (Q × Q)) | |
3 | 2 | breqi 5154 | . . 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 5696 | . . . . 5 ⊢ Rel (N × N) | |
6 | elpqn 10949 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
7 | 1st2nd 8043 | . . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
8 | 5, 6, 7 | sylancr 586 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
9 | elpqn 10949 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
10 | 1st2nd 8043 | . . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
11 | 5, 9, 10 | sylancr 586 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) |
12 | 8, 11 | breqan12d 5164 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 <pQ 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
13 | ordpipq 10966 | . . 3 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 <pQ 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))) | |
14 | 12, 13 | bitrdi 287 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
15 | 4, 14 | bitr3d 281 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 〈cop 4635 class class class wbr 5148 × cxp 5676 Rel wrel 5683 ‘cfv 6548 (class class class)co 7420 1st c1st 7991 2nd c2nd 7992 Ncnpi 10868 ·N cmi 10870 <N clti 10871 <pQ cltpq 10874 Qcnq 10876 <Q cltq 10882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-omul 8492 df-ni 10896 df-mi 10898 df-lti 10899 df-ltpq 10934 df-nq 10936 df-ltnq 10942 |
This theorem is referenced by: ltsonq 10993 lterpq 10994 ltanq 10995 ltmnq 10996 ltexnq 10999 archnq 11004 |
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