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Mirrors > Home > ILE Home > Th. List > divnumden | Unicode version |
Description: Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
divnumden | numer denom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 | |
2 | nnz 9073 | . . . . 5 | |
3 | 2 | adantl 275 | . . . 4 |
4 | nnne0 8748 | . . . . . . . 8 | |
5 | 4 | neneqd 2329 | . . . . . . 7 |
6 | 5 | adantl 275 | . . . . . 6 |
7 | 6 | intnand 916 | . . . . 5 |
8 | gcdn0cl 11651 | . . . . 5 | |
9 | 1, 3, 7, 8 | syl21anc 1215 | . . . 4 |
10 | gcddvds 11652 | . . . . 5 | |
11 | 2, 10 | sylan2 284 | . . . 4 |
12 | gcddiv 11707 | . . . 4 | |
13 | 1, 3, 9, 11, 12 | syl31anc 1219 | . . 3 |
14 | 9 | nncnd 8734 | . . . 4 |
15 | 9 | nnap0d 8766 | . . . 4 # |
16 | 14, 15 | dividapd 8546 | . . 3 |
17 | 13, 16 | eqtr3d 2174 | . 2 |
18 | zcn 9059 | . . . 4 | |
19 | 18 | adantr 274 | . . 3 |
20 | nncn 8728 | . . . 4 | |
21 | 20 | adantl 275 | . . 3 |
22 | simpr 109 | . . . 4 | |
23 | 22 | nnap0d 8766 | . . 3 # |
24 | divcanap7 8481 | . . . 4 # # | |
25 | 24 | eqcomd 2145 | . . 3 # # |
26 | 19, 21, 23, 14, 15, 25 | syl122anc 1225 | . 2 |
27 | znq 9416 | . . 3 | |
28 | 11 | simpld 111 | . . . 4 |
29 | gcdcl 11655 | . . . . . . 7 | |
30 | 29 | nn0zd 9171 | . . . . . 6 |
31 | 2, 30 | sylan2 284 | . . . . 5 |
32 | 9 | nnne0d 8765 | . . . . 5 |
33 | dvdsval2 11496 | . . . . 5 | |
34 | 31, 32, 1, 33 | syl3anc 1216 | . . . 4 |
35 | 28, 34 | mpbid 146 | . . 3 |
36 | 11 | simprd 113 | . . . 4 |
37 | nndivdvds 11499 | . . . . 5 | |
38 | 22, 9, 37 | syl2anc 408 | . . . 4 |
39 | 36, 38 | mpbid 146 | . . 3 |
40 | qnumdenbi 11870 | . . 3 numer denom | |
41 | 27, 35, 39, 40 | syl3anc 1216 | . 2 numer denom |
42 | 17, 26, 41 | mpbi2and 927 | 1 numer denom |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cc0 7620 c1 7621 # cap 8343 cdiv 8432 cn 8720 cz 9054 cq 9411 cdvds 11493 cgcd 11635 numercnumer 11859 denomcdenom 11860 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-fl 10043 df-mod 10096 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-dvds 11494 df-gcd 11636 df-numer 11861 df-denom 11862 |
This theorem is referenced by: divdenle 11875 |
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