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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 4621 | . . . 4 | |
2 | 1 | 3adant1 1000 | . . 3 |
3 | qredeu 11990 | . . . 4 | |
4 | 3 | 3ad2ant1 1003 | . . 3 |
5 | fveq2 5471 | . . . . . . 7 | |
6 | fveq2 5471 | . . . . . . 7 | |
7 | 5, 6 | oveq12d 5845 | . . . . . 6 |
8 | 7 | eqeq1d 2166 | . . . . 5 |
9 | 5, 6 | oveq12d 5845 | . . . . . 6 |
10 | 9 | eqeq2d 2169 | . . . . 5 |
11 | 8, 10 | anbi12d 465 | . . . 4 |
12 | 11 | riota2 5805 | . . 3 |
13 | 2, 4, 12 | syl2anc 409 | . 2 |
14 | op1stg 6101 | . . . . . 6 | |
15 | op2ndg 6102 | . . . . . 6 | |
16 | 14, 15 | oveq12d 5845 | . . . . 5 |
17 | 16 | 3adant1 1000 | . . . 4 |
18 | 17 | eqeq1d 2166 | . . 3 |
19 | 14 | 3adant1 1000 | . . . . 5 |
20 | 15 | 3adant1 1000 | . . . . 5 |
21 | 19, 20 | oveq12d 5845 | . . . 4 |
22 | 21 | eqeq2d 2169 | . . 3 |
23 | 18, 22 | anbi12d 465 | . 2 |
24 | riotacl 5797 | . . . . . . 7 | |
25 | 1st2nd2 6126 | . . . . . . 7 | |
26 | 3, 24, 25 | 3syl 17 | . . . . . 6 |
27 | qnumval 12076 | . . . . . . 7 numer | |
28 | qdenval 12077 | . . . . . . 7 denom | |
29 | 27, 28 | opeq12d 3751 | . . . . . 6 numer denom |
30 | 26, 29 | eqtr4d 2193 | . . . . 5 numer denom |
31 | 30 | eqeq1d 2166 | . . . 4 numer denom |
32 | 31 | 3ad2ant1 1003 | . . 3 numer denom |
33 | qnumcl 12079 | . . . . 5 numer | |
34 | qdencl 12080 | . . . . 5 denom | |
35 | opthg 4201 | . . . . 5 numer denom numer denom numer denom | |
36 | 33, 34, 35 | syl2anc 409 | . . . 4 numer denom numer denom |
37 | 36 | 3ad2ant1 1003 | . . 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 963 wceq 1335 wcel 2128 wreu 2437 cop 3564 cxp 4587 cfv 5173 crio 5782 (class class class)co 5827 c1st 6089 c2nd 6090 c1 7736 cdiv 8550 cn 8839 cz 9173 cq 9535 cgcd 11842 numercnumer 12072 denomcdenom 12073 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 ax-arch 7854 ax-caucvg 7855 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-frec 6341 df-sup 6931 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-2 8898 df-3 8899 df-4 8900 df-n0 9097 df-z 9174 df-uz 9446 df-q 9536 df-rp 9568 df-fz 9920 df-fzo 10052 df-fl 10179 df-mod 10232 df-seqfrec 10355 df-exp 10429 df-cj 10754 df-re 10755 df-im 10756 df-rsqrt 10910 df-abs 10911 df-dvds 11696 df-gcd 11843 df-numer 12074 df-denom 12075 |
This theorem is referenced by: qnumdencoprm 12084 qeqnumdivden 12085 divnumden 12087 numdensq 12093 |
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