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Mirrors > Home > ILE Home > Th. List > intfracq | Unicode version |
Description: Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10218. (Contributed by NM, 16-Aug-2008.) |
Ref | Expression |
---|---|
intfracq.1 | |
intfracq.2 |
Ref | Expression |
---|---|
intfracq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znq 9533 | . . . 4 | |
2 | intfracq.1 | . . . . 5 | |
3 | intfracq.2 | . . . . 5 | |
4 | 2, 3 | intqfrac2 10218 | . . . 4 |
5 | 1, 4 | syl 14 | . . 3 |
6 | 5 | simp1d 994 | . 2 |
7 | qfraclt1 10179 | . . . . . . 7 | |
8 | 1, 7 | syl 14 | . . . . . 6 |
9 | 2 | oveq2i 5835 | . . . . . . . 8 |
10 | 3, 9 | eqtri 2178 | . . . . . . 7 |
11 | 10 | a1i 9 | . . . . . 6 |
12 | simpr 109 | . . . . . . . 8 | |
13 | 12 | nncnd 8847 | . . . . . . 7 |
14 | 12 | nnap0d 8879 | . . . . . . 7 # |
15 | 13, 14 | dividapd 8659 | . . . . . 6 |
16 | 8, 11, 15 | 3brtr4d 3996 | . . . . 5 |
17 | qre 9534 | . . . . . . . . 9 | |
18 | 1, 17 | syl 14 | . . . . . . . 8 |
19 | 1 | flqcld 10176 | . . . . . . . . . 10 |
20 | 2, 19 | eqeltrid 2244 | . . . . . . . . 9 |
21 | 20 | zred 9286 | . . . . . . . 8 |
22 | 18, 21 | resubcld 8256 | . . . . . . 7 |
23 | 3, 22 | eqeltrid 2244 | . . . . . 6 |
24 | nnre 8840 | . . . . . . 7 | |
25 | 24 | adantl 275 | . . . . . 6 |
26 | nngt0 8858 | . . . . . . . 8 | |
27 | 24, 26 | jca 304 | . . . . . . 7 |
28 | 27 | adantl 275 | . . . . . 6 |
29 | ltmuldiv2 8746 | . . . . . 6 | |
30 | 23, 25, 28, 29 | syl3anc 1220 | . . . . 5 |
31 | 16, 30 | mpbird 166 | . . . 4 |
32 | 3 | oveq2i 5835 | . . . . . . 7 |
33 | 18 | recnd 7906 | . . . . . . . 8 |
34 | 20 | zcnd 9287 | . . . . . . . 8 |
35 | 13, 33, 34 | subdid 8289 | . . . . . . 7 |
36 | 32, 35 | syl5eq 2202 | . . . . . 6 |
37 | zcn 9172 | . . . . . . . . . 10 | |
38 | 37 | adantr 274 | . . . . . . . . 9 |
39 | 38, 13, 14 | divcanap2d 8665 | . . . . . . . 8 |
40 | simpl 108 | . . . . . . . 8 | |
41 | 39, 40 | eqeltrd 2234 | . . . . . . 7 |
42 | nnz 9186 | . . . . . . . . 9 | |
43 | 42 | adantl 275 | . . . . . . . 8 |
44 | 43, 20 | zmulcld 9292 | . . . . . . 7 |
45 | 41, 44 | zsubcld 9291 | . . . . . 6 |
46 | 36, 45 | eqeltrd 2234 | . . . . 5 |
47 | zltlem1 9224 | . . . . 5 | |
48 | 46, 43, 47 | syl2anc 409 | . . . 4 |
49 | 31, 48 | mpbid 146 | . . 3 |
50 | peano2rem 8142 | . . . . . 6 | |
51 | 24, 50 | syl 14 | . . . . 5 |
52 | 51 | adantl 275 | . . . 4 |
53 | lemuldiv2 8753 | . . . 4 | |
54 | 23, 52, 28, 53 | syl3anc 1220 | . . 3 |
55 | 49, 54 | mpbid 146 | . 2 |
56 | 5 | simp3d 996 | . 2 |
57 | 6, 55, 56 | 3jca 1162 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 class class class wbr 3965 cfv 5170 (class class class)co 5824 cc 7730 cr 7731 cc0 7732 c1 7733 caddc 7735 cmul 7737 clt 7912 cle 7913 cmin 8046 cdiv 8545 cn 8833 cz 9167 cq 9528 cfl 10167 |
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-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 |
This theorem depends on definitions: df-bi 116 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-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-po 4256 df-iso 4257 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-n0 9091 df-z 9168 df-q 9529 df-rp 9561 df-fl 10169 |
This theorem is referenced by: flqdiv 10220 |
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