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Mirrors > Home > ILE Home > Th. List > qnumdenbi | Unicode version |
Description: Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
qnumdenbi | numer denom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4541 | . . . 4 | |
2 | 1 | 3adant1 984 | . . 3 |
3 | qredeu 11705 | . . . 4 | |
4 | 3 | 3ad2ant1 987 | . . 3 |
5 | fveq2 5389 | . . . . . . 7 | |
6 | fveq2 5389 | . . . . . . 7 | |
7 | 5, 6 | oveq12d 5760 | . . . . . 6 |
8 | 7 | eqeq1d 2126 | . . . . 5 |
9 | 5, 6 | oveq12d 5760 | . . . . . 6 |
10 | 9 | eqeq2d 2129 | . . . . 5 |
11 | 8, 10 | anbi12d 464 | . . . 4 |
12 | 11 | riota2 5720 | . . 3 |
13 | 2, 4, 12 | syl2anc 408 | . 2 |
14 | op1stg 6016 | . . . . . 6 | |
15 | op2ndg 6017 | . . . . . 6 | |
16 | 14, 15 | oveq12d 5760 | . . . . 5 |
17 | 16 | 3adant1 984 | . . . 4 |
18 | 17 | eqeq1d 2126 | . . 3 |
19 | 14 | 3adant1 984 | . . . . 5 |
20 | 15 | 3adant1 984 | . . . . 5 |
21 | 19, 20 | oveq12d 5760 | . . . 4 |
22 | 21 | eqeq2d 2129 | . . 3 |
23 | 18, 22 | anbi12d 464 | . 2 |
24 | riotacl 5712 | . . . . . . 7 | |
25 | 1st2nd2 6041 | . . . . . . 7 | |
26 | 3, 24, 25 | 3syl 17 | . . . . . 6 |
27 | qnumval 11790 | . . . . . . 7 numer | |
28 | qdenval 11791 | . . . . . . 7 denom | |
29 | 27, 28 | opeq12d 3683 | . . . . . 6 numer denom |
30 | 26, 29 | eqtr4d 2153 | . . . . 5 numer denom |
31 | 30 | eqeq1d 2126 | . . . 4 numer denom |
32 | 31 | 3ad2ant1 987 | . . 3 numer denom |
33 | qnumcl 11793 | . . . . 5 numer | |
34 | qdencl 11794 | . . . . 5 denom | |
35 | opthg 4130 | . . . . 5 numer denom numer denom numer denom | |
36 | 33, 34, 35 | syl2anc 408 | . . . 4 numer denom numer denom |
37 | 36 | 3ad2ant1 987 | . . 3 numer denom numer denom |
38 | 32, 37 | bitrd 187 | . 2 numer denom |
39 | 13, 23, 38 | 3bitr3d 217 | 1 numer denom |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wreu 2395 cop 3500 cxp 4507 cfv 5093 crio 5697 (class class class)co 5742 c1st 6004 c2nd 6005 c1 7589 cdiv 8400 cn 8688 cz 9022 cq 9379 cgcd 11562 numercnumer 11786 denomcdenom 11787 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-sup 6839 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 df-q 9380 df-rp 9410 df-fz 9759 df-fzo 9888 df-fl 10011 df-mod 10064 df-seqfrec 10187 df-exp 10261 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 df-dvds 11421 df-gcd 11563 df-numer 11788 df-denom 11789 |
This theorem is referenced by: qnumdencoprm 11798 qeqnumdivden 11799 divnumden 11801 numdensq 11807 |
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