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Mirrors > Home > MPE 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 intfrac2 11985. (Contributed by NM, 16-Aug-2008.) |
Ref | Expression |
---|---|
intfracq.1 | |
intfracq.2 |
Ref | Expression |
---|---|
intfracq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 10893 | . . . . . 6 | |
2 | 1 | adantr 465 | . . . . 5 |
3 | nnre 10568 | . . . . . 6 | |
4 | 3 | adantl 466 | . . . . 5 |
5 | nnne0 10593 | . . . . . 6 | |
6 | 5 | adantl 466 | . . . . 5 |
7 | 2, 4, 6 | redivcld 10397 | . . . 4 |
8 | intfracq.1 | . . . . 5 | |
9 | intfracq.2 | . . . . 5 | |
10 | 8, 9 | intfrac2 11985 | . . . 4 |
11 | 7, 10 | syl 16 | . . 3 |
12 | 11 | simp1d 1008 | . 2 |
13 | fraclt1 11939 | . . . . . . 7 | |
14 | 7, 13 | syl 16 | . . . . . 6 |
15 | 8 | oveq2i 6307 | . . . . . . . 8 |
16 | 9, 15 | eqtri 2486 | . . . . . . 7 |
17 | 16 | a1i 11 | . . . . . 6 |
18 | nncn 10569 | . . . . . . . 8 | |
19 | 18, 5 | dividd 10343 | . . . . . . 7 |
20 | 19 | adantl 466 | . . . . . 6 |
21 | 14, 17, 20 | 3brtr4d 4482 | . . . . 5 |
22 | reflcl 11933 | . . . . . . . . . 10 | |
23 | 7, 22 | syl 16 | . . . . . . . . 9 |
24 | 8, 23 | syl5eqel 2549 | . . . . . . . 8 |
25 | 7, 24 | resubcld 10012 | . . . . . . 7 |
26 | 9, 25 | syl5eqel 2549 | . . . . . 6 |
27 | nngt0 10590 | . . . . . . . 8 | |
28 | 3, 27 | jca 532 | . . . . . . 7 |
29 | 28 | adantl 466 | . . . . . 6 |
30 | ltmuldiv2 10441 | . . . . . 6 | |
31 | 26, 4, 29, 30 | syl3anc 1228 | . . . . 5 |
32 | 21, 31 | mpbird 232 | . . . 4 |
33 | 9 | oveq2i 6307 | . . . . . . 7 |
34 | 18 | adantl 466 | . . . . . . . 8 |
35 | 7 | recnd 9643 | . . . . . . . 8 |
36 | 7 | flcld 11935 | . . . . . . . . . 10 |
37 | 8, 36 | syl5eqel 2549 | . . . . . . . . 9 |
38 | 37 | zcnd 10995 | . . . . . . . 8 |
39 | 34, 35, 38 | subdid 10037 | . . . . . . 7 |
40 | 33, 39 | syl5eq 2510 | . . . . . 6 |
41 | zcn 10894 | . . . . . . . . . 10 | |
42 | 41 | adantr 465 | . . . . . . . . 9 |
43 | 42, 34, 6 | divcan2d 10347 | . . . . . . . 8 |
44 | simpl 457 | . . . . . . . 8 | |
45 | 43, 44 | eqeltrd 2545 | . . . . . . 7 |
46 | nnz 10911 | . . . . . . . . 9 | |
47 | 46 | adantl 466 | . . . . . . . 8 |
48 | 47, 37 | zmulcld 11000 | . . . . . . 7 |
49 | 45, 48 | zsubcld 10999 | . . . . . 6 |
50 | 40, 49 | eqeltrd 2545 | . . . . 5 |
51 | zltlem1 10941 | . . . . 5 | |
52 | 50, 47, 51 | syl2anc 661 | . . . 4 |
53 | 32, 52 | mpbid 210 | . . 3 |
54 | peano2rem 9909 | . . . . . 6 | |
55 | 3, 54 | syl 16 | . . . . 5 |
56 | 55 | adantl 466 | . . . 4 |
57 | lemuldiv2 10450 | . . . 4 | |
58 | 26, 56, 29, 57 | syl3anc 1228 | . . 3 |
59 | 53, 58 | mpbid 210 | . 2 |
60 | 11 | simp3d 1010 | . 2 |
61 | 12, 59, 60 | 3jca 1176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 /\ w3a 973 = wceq 1395
e. wcel 1818 =/= wne 2652 class class class wbr 4452
` cfv 5593 (class class class)co 6296
cc 9511 cr 9512 0 cc0 9513 1 c1 9514
caddc 9516 cmul 9518 clt 9649 cle 9650 cmin 9828 cdiv 10231 cn 10561 cz 10889 cfl 11927 |
This theorem is referenced by: fldiv 11987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-recs 7061 df-rdg 7095 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-sup 7921 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 df-nn 10562 df-n0 10821 df-z 10890 df-uz 11111 df-fl 11929 |
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