Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5623 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵)) | |
2 | df-ltnq 10328 | . . . 4 ⊢ <Q = ( <pQ ∩ (Q × Q)) | |
3 | 2 | breqi 5063 | . . 3 ⊢ (𝐴 <Q 𝐵 ↔ 𝐴( <pQ ∩ (Q × Q))𝐵) |
4 | 1, 3 | syl6bbr 290 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 𝐴 <Q 𝐵)) |
5 | relxp 5566 | . . . . 5 ⊢ Rel (N × N) | |
6 | elpqn 10335 | . . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
7 | 1st2nd 7727 | . . . . 5 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
8 | 5, 6, 7 | sylancr 587 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
9 | elpqn 10335 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
10 | 1st2nd 7727 | . . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
11 | 5, 9, 10 | sylancr 587 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) |
12 | 8, 11 | breqan12d 5073 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 <pQ 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
13 | ordpipq 10352 | . . 3 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 <pQ 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))) | |
14 | 12, 13 | syl6bb 288 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
15 | 4, 14 | bitr3d 282 | 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 〈cop 4563 class class class wbr 5057 × cxp 5546 Rel wrel 5553 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 Ncnpi 10254 ·N cmi 10256 <N clti 10257 <pQ cltpq 10260 Qcnq 10262 <Q cltq 10268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-omul 8096 df-ni 10282 df-mi 10284 df-lti 10285 df-ltpq 10320 df-nq 10322 df-ltnq 10328 |
This theorem is referenced by: ltsonq 10379 lterpq 10380 ltanq 10381 ltmnq 10382 ltexnq 10385 archnq 10390 |
Copyright terms: Public domain | W3C validator |