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Mirrors > Home > ILE Home > Th. List > qnumgt0 | GIF version |
Description: A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Ref | Expression |
---|---|
qnumgt0 | β’ (π΄ β β β (0 < π΄ β 0 < (numerβπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 7933 | . . 3 β’ (π΄ β β β 0 β β) | |
2 | qre 9596 | . . 3 β’ (π΄ β β β π΄ β β) | |
3 | qdencl 12155 | . . . 4 β’ (π΄ β β β (denomβπ΄) β β) | |
4 | 3 | nnred 8903 | . . 3 β’ (π΄ β β β (denomβπ΄) β β) |
5 | 3 | nngt0d 8934 | . . 3 β’ (π΄ β β β 0 < (denomβπ΄)) |
6 | ltmul1 8523 | . . 3 β’ ((0 β β β§ π΄ β β β§ ((denomβπ΄) β β β§ 0 < (denomβπ΄))) β (0 < π΄ β (0 Β· (denomβπ΄)) < (π΄ Β· (denomβπ΄)))) | |
7 | 1, 2, 4, 5, 6 | syl112anc 1242 | . 2 β’ (π΄ β β β (0 < π΄ β (0 Β· (denomβπ΄)) < (π΄ Β· (denomβπ΄)))) |
8 | 3 | nncnd 8904 | . . . 4 β’ (π΄ β β β (denomβπ΄) β β) |
9 | 8 | mul02d 8323 | . . 3 β’ (π΄ β β β (0 Β· (denomβπ΄)) = 0) |
10 | qmuldeneqnum 12161 | . . 3 β’ (π΄ β β β (π΄ Β· (denomβπ΄)) = (numerβπ΄)) | |
11 | 9, 10 | breq12d 4011 | . 2 β’ (π΄ β β β ((0 Β· (denomβπ΄)) < (π΄ Β· (denomβπ΄)) β 0 < (numerβπ΄))) |
12 | 7, 11 | bitrd 188 | 1 β’ (π΄ β β β (0 < π΄ β 0 < (numerβπ΄))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wb 105 β wcel 2146 class class class wbr 3998 βcfv 5208 (class class class)co 5865 βcr 7785 0cc0 7786 Β· cmul 7791 < clt 7966 βcq 9590 numercnumer 12147 denomcdenom 12148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-sup 6973 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-q 9591 df-rp 9623 df-fz 9978 df-fzo 10111 df-fl 10238 df-mod 10291 df-seqfrec 10414 df-exp 10488 df-cj 10818 df-re 10819 df-im 10820 df-rsqrt 10974 df-abs 10975 df-dvds 11762 df-gcd 11910 df-numer 12149 df-denom 12150 |
This theorem is referenced by: qgt0numnn 12165 |
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