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Mirrors > Home > ILE Home > Th. List > ltrnqi | GIF version |
Description: Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6881. (Contributed by Jim Kingdon, 24-Sep-2019.) |
Ref | Expression |
---|---|
ltrnqi | ⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 6826 | . . . 4 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4447 | . . 3 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q)) |
3 | ltrnqg 6881 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝐴 <Q 𝐵 → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))) |
5 | 4 | ibi 174 | 1 ⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1434 class class class wbr 3811 ‘cfv 4968 Qcnq 6741 *Qcrq 6745 <Q cltq 6746 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-eprel 4079 df-id 4083 df-iord 4156 df-on 4158 df-suc 4161 df-iom 4368 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-recs 6001 df-irdg 6066 df-1o 6112 df-oadd 6116 df-omul 6117 df-er 6221 df-ec 6223 df-qs 6227 df-ni 6765 df-mi 6767 df-lti 6768 df-mpq 6806 df-enq 6808 df-nqqs 6809 df-mqqs 6811 df-1nqqs 6812 df-rq 6813 df-ltnqqs 6814 |
This theorem is referenced by: addnqprllem 6988 addnqprulem 6989 recexprlemdisj 7091 recexprlemloc 7092 recexprlem1ssl 7094 recexprlem1ssu 7095 caucvgprlemk 7126 caucvgprprlemk 7144 |
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