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Mirrors > Home > ILE Home > Th. List > qnumdencoprm | Unicode version |
Description: The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
qnumdencoprm | numer denom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2165 | . . . 4 numer numer | |
2 | eqid 2164 | . . . 4 denom denom | |
3 | 1, 2 | jctir 311 | . . 3 numer numer denom denom |
4 | qnumcl 12109 | . . . 4 numer | |
5 | qdencl 12110 | . . . 4 denom | |
6 | qnumdenbi 12113 | . . . 4 numer denom numer denom numer denom numer numer denom denom | |
7 | 4, 5, 6 | mpd3an23 1328 | . . 3 numer denom numer denom numer numer denom denom |
8 | 3, 7 | mpbird 166 | . 2 numer denom numer denom |
9 | 8 | simpld 111 | 1 numer denom |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cfv 5183 (class class class)co 5837 c1 7746 cdiv 8560 cn 8849 cz 9183 cq 9549 cgcd 11864 numercnumer 12102 denomcdenom 12103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 ax-arch 7864 ax-caucvg 7865 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-frec 6351 df-sup 6941 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 df-inn 8850 df-2 8908 df-3 8909 df-4 8910 df-n0 9107 df-z 9184 df-uz 9459 df-q 9550 df-rp 9582 df-fz 9937 df-fzo 10069 df-fl 10196 df-mod 10249 df-seqfrec 10372 df-exp 10446 df-cj 10774 df-re 10775 df-im 10776 df-rsqrt 10930 df-abs 10931 df-dvds 11718 df-gcd 11865 df-numer 12104 df-denom 12105 |
This theorem is referenced by: numdensq 12123 |
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