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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 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 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
8 | 5, 6, 7 | sylancr 581 | . . . 4 ⊢ (𝐴 ∈ Q → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
9 | elpqn 10082 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
10 | 1st2nd 7493 | . . . . 5 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
11 | 5, 9, 10 | sylancr 581 | . . . 4 ⊢ (𝐵 ∈ Q → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) |
12 | 8, 11 | breqan12d 4902 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 <pQ 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
13 | ordpipq 10099 | . . 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 279 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <pQ 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) |
15 | 4, 14 | bitr3d 273 | 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 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 1st c1st 7443 2nd 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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