Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > flqdiv | Unicode version |
Description: Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
flqdiv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2140 | . . . . . . . . 9 | |
2 | eqid 2140 | . . . . . . . . 9 | |
3 | 1, 2 | intqfrac2 10123 | . . . . . . . 8 |
4 | 3 | simp3d 996 | . . . . . . 7 |
5 | 4 | adantr 274 | . . . . . 6 |
6 | 5 | oveq1d 5797 | . . . . 5 |
7 | simpl 108 | . . . . . . . 8 | |
8 | 7 | flqcld 10081 | . . . . . . 7 |
9 | 8 | zcnd 9198 | . . . . . 6 |
10 | zq 9445 | . . . . . . . 8 | |
11 | 8, 10 | syl 14 | . . . . . . 7 |
12 | qsubcl 9457 | . . . . . . . 8 | |
13 | qcn 9453 | . . . . . . . 8 | |
14 | 12, 13 | syl 14 | . . . . . . 7 |
15 | 11, 14 | syldan 280 | . . . . . 6 |
16 | simpr 109 | . . . . . . 7 | |
17 | 16 | nncnd 8758 | . . . . . 6 |
18 | 16 | nnap0d 8790 | . . . . . 6 # |
19 | 9, 15, 17, 18 | divdirapd 8613 | . . . . 5 |
20 | 6, 19 | eqtrd 2173 | . . . 4 |
21 | flqcl 10077 | . . . . . 6 | |
22 | eqid 2140 | . . . . . . . 8 | |
23 | eqid 2140 | . . . . . . . 8 | |
24 | 22, 23 | intfracq 10124 | . . . . . . 7 |
25 | 24 | simp3d 996 | . . . . . 6 |
26 | 21, 25 | sylan 281 | . . . . 5 |
27 | 26 | oveq1d 5797 | . . . 4 |
28 | znq 9443 | . . . . . . . 8 | |
29 | 28 | flqcld 10081 | . . . . . . 7 |
30 | 21, 29 | sylan 281 | . . . . . 6 |
31 | 30 | zcnd 9198 | . . . . 5 |
32 | 8, 16, 28 | syl2anc 409 | . . . . . . 7 |
33 | zq 9445 | . . . . . . . 8 | |
34 | 30, 33 | syl 14 | . . . . . . 7 |
35 | qsubcl 9457 | . . . . . . 7 | |
36 | 32, 34, 35 | syl2anc 409 | . . . . . 6 |
37 | qcn 9453 | . . . . . 6 | |
38 | 36, 37 | syl 14 | . . . . 5 |
39 | 11, 12 | syldan 280 | . . . . . . 7 |
40 | nnq 9452 | . . . . . . . 8 | |
41 | 40 | adantl 275 | . . . . . . 7 |
42 | 16 | nnne0d 8789 | . . . . . . 7 |
43 | qdivcl 9462 | . . . . . . 7 | |
44 | 39, 41, 42, 43 | syl3anc 1217 | . . . . . 6 |
45 | qcn 9453 | . . . . . 6 | |
46 | 44, 45 | syl 14 | . . . . 5 |
47 | 31, 38, 46 | addassd 7812 | . . . 4 |
48 | 20, 27, 47 | 3eqtrd 2177 | . . 3 |
49 | 48 | fveq2d 5433 | . 2 |
50 | qre 9444 | . . . . 5 | |
51 | 36, 50 | syl 14 | . . . 4 |
52 | qre 9444 | . . . . . 6 | |
53 | 39, 52 | syl 14 | . . . . 5 |
54 | 53, 16 | nndivred 8794 | . . . 4 |
55 | 24 | simp1d 994 | . . . . 5 |
56 | 21, 55 | sylan 281 | . . . 4 |
57 | 16 | nnrpd 9511 | . . . . 5 |
58 | qfracge0 10085 | . . . . . 6 | |
59 | 58 | adantr 274 | . . . . 5 |
60 | 53, 57, 59 | divge0d 9554 | . . . 4 |
61 | 51, 54, 56, 60 | addge0d 8308 | . . 3 |
62 | nnre 8751 | . . . . . . . 8 | |
63 | peano2rem 8053 | . . . . . . . 8 | |
64 | 62, 63 | syl 14 | . . . . . . 7 |
65 | nnap0 8773 | . . . . . . 7 # | |
66 | 64, 62, 65 | redivclapd 8618 | . . . . . 6 |
67 | 66 | adantl 275 | . . . . 5 |
68 | 16 | nnrecred 8791 | . . . . 5 |
69 | 24 | simp2d 995 | . . . . . 6 |
70 | 21, 69 | sylan 281 | . . . . 5 |
71 | qfraclt1 10084 | . . . . . . 7 | |
72 | 71 | adantr 274 | . . . . . 6 |
73 | 16 | nnred 8757 | . . . . . . 7 |
74 | 16 | nngt0d 8788 | . . . . . . 7 |
75 | 1re 7789 | . . . . . . . 8 | |
76 | ltdiv1 8650 | . . . . . . . 8 | |
77 | 75, 76 | mp3an2 1304 | . . . . . . 7 |
78 | 53, 73, 74, 77 | syl12anc 1215 | . . . . . 6 |
79 | 72, 78 | mpbid 146 | . . . . 5 |
80 | 51, 54, 67, 68, 70, 79 | leltaddd 8352 | . . . 4 |
81 | nncn 8752 | . . . . . . . 8 | |
82 | npcan1 8164 | . . . . . . . 8 | |
83 | 81, 82 | syl 14 | . . . . . . 7 |
84 | 83 | oveq1d 5797 | . . . . . 6 |
85 | 64 | recnd 7818 | . . . . . . 7 |
86 | ax-1cn 7737 | . . . . . . . 8 | |
87 | divdirap 8481 | . . . . . . . 8 # | |
88 | 86, 87 | mp3an2 1304 | . . . . . . 7 # |
89 | 85, 81, 65, 88 | syl12anc 1215 | . . . . . 6 |
90 | 81, 65 | dividapd 8570 | . . . . . 6 |
91 | 84, 89, 90 | 3eqtr3d 2181 | . . . . 5 |
92 | 91 | adantl 275 | . . . 4 |
93 | 80, 92 | breqtrd 3962 | . . 3 |
94 | 32 | flqcld 10081 | . . . 4 |
95 | qaddcl 9454 | . . . . 5 | |
96 | 36, 44, 95 | syl2anc 409 | . . . 4 |
97 | flqbi2 10095 | . . . 4 | |
98 | 94, 96, 97 | syl2anc 409 | . . 3 |
99 | 61, 93, 98 | mpbir2and 929 | . 2 |
100 | 49, 99 | eqtr2d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1332 wcel 1481 wne 2309 class class class wbr 3937 cfv 5131 (class class class)co 5782 cc 7642 cr 7643 cc0 7644 c1 7645 caddc 7647 clt 7824 cle 7825 cmin 7957 # cap 8367 cdiv 8456 cn 8744 cz 9078 cq 9438 cfl 10072 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-n0 9002 df-z 9079 df-q 9439 df-rp 9471 df-fl 10074 |
This theorem is referenced by: modqmulnn 10146 |
Copyright terms: Public domain | W3C validator |