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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 2139 | . . . . . . . . 9 | |
2 | eqid 2139 | . . . . . . . . 9 | |
3 | 1, 2 | intqfrac2 10092 | . . . . . . . 8 |
4 | 3 | simp3d 995 | . . . . . . 7 |
5 | 4 | adantr 274 | . . . . . 6 |
6 | 5 | oveq1d 5789 | . . . . 5 |
7 | simpl 108 | . . . . . . . 8 | |
8 | 7 | flqcld 10050 | . . . . . . 7 |
9 | 8 | zcnd 9174 | . . . . . 6 |
10 | zq 9418 | . . . . . . . 8 | |
11 | 8, 10 | syl 14 | . . . . . . 7 |
12 | qsubcl 9430 | . . . . . . . 8 | |
13 | qcn 9426 | . . . . . . . 8 | |
14 | 12, 13 | syl 14 | . . . . . . 7 |
15 | 11, 14 | syldan 280 | . . . . . 6 |
16 | simpr 109 | . . . . . . 7 | |
17 | 16 | nncnd 8734 | . . . . . 6 |
18 | 16 | nnap0d 8766 | . . . . . 6 # |
19 | 9, 15, 17, 18 | divdirapd 8589 | . . . . 5 |
20 | 6, 19 | eqtrd 2172 | . . . 4 |
21 | flqcl 10046 | . . . . . 6 | |
22 | eqid 2139 | . . . . . . . 8 | |
23 | eqid 2139 | . . . . . . . 8 | |
24 | 22, 23 | intfracq 10093 | . . . . . . 7 |
25 | 24 | simp3d 995 | . . . . . 6 |
26 | 21, 25 | sylan 281 | . . . . 5 |
27 | 26 | oveq1d 5789 | . . . 4 |
28 | znq 9416 | . . . . . . . 8 | |
29 | 28 | flqcld 10050 | . . . . . . 7 |
30 | 21, 29 | sylan 281 | . . . . . 6 |
31 | 30 | zcnd 9174 | . . . . 5 |
32 | 8, 16, 28 | syl2anc 408 | . . . . . . 7 |
33 | zq 9418 | . . . . . . . 8 | |
34 | 30, 33 | syl 14 | . . . . . . 7 |
35 | qsubcl 9430 | . . . . . . 7 | |
36 | 32, 34, 35 | syl2anc 408 | . . . . . 6 |
37 | qcn 9426 | . . . . . 6 | |
38 | 36, 37 | syl 14 | . . . . 5 |
39 | 11, 12 | syldan 280 | . . . . . . 7 |
40 | nnq 9425 | . . . . . . . 8 | |
41 | 40 | adantl 275 | . . . . . . 7 |
42 | 16 | nnne0d 8765 | . . . . . . 7 |
43 | qdivcl 9435 | . . . . . . 7 | |
44 | 39, 41, 42, 43 | syl3anc 1216 | . . . . . 6 |
45 | qcn 9426 | . . . . . 6 | |
46 | 44, 45 | syl 14 | . . . . 5 |
47 | 31, 38, 46 | addassd 7788 | . . . 4 |
48 | 20, 27, 47 | 3eqtrd 2176 | . . 3 |
49 | 48 | fveq2d 5425 | . 2 |
50 | qre 9417 | . . . . 5 | |
51 | 36, 50 | syl 14 | . . . 4 |
52 | qre 9417 | . . . . . 6 | |
53 | 39, 52 | syl 14 | . . . . 5 |
54 | 53, 16 | nndivred 8770 | . . . 4 |
55 | 24 | simp1d 993 | . . . . 5 |
56 | 21, 55 | sylan 281 | . . . 4 |
57 | 16 | nnrpd 9482 | . . . . 5 |
58 | qfracge0 10054 | . . . . . 6 | |
59 | 58 | adantr 274 | . . . . 5 |
60 | 53, 57, 59 | divge0d 9524 | . . . 4 |
61 | 51, 54, 56, 60 | addge0d 8284 | . . 3 |
62 | nnre 8727 | . . . . . . . 8 | |
63 | peano2rem 8029 | . . . . . . . 8 | |
64 | 62, 63 | syl 14 | . . . . . . 7 |
65 | nnap0 8749 | . . . . . . 7 # | |
66 | 64, 62, 65 | redivclapd 8594 | . . . . . 6 |
67 | 66 | adantl 275 | . . . . 5 |
68 | 16 | nnrecred 8767 | . . . . 5 |
69 | 24 | simp2d 994 | . . . . . 6 |
70 | 21, 69 | sylan 281 | . . . . 5 |
71 | qfraclt1 10053 | . . . . . . 7 | |
72 | 71 | adantr 274 | . . . . . 6 |
73 | 16 | nnred 8733 | . . . . . . 7 |
74 | 16 | nngt0d 8764 | . . . . . . 7 |
75 | 1re 7765 | . . . . . . . 8 | |
76 | ltdiv1 8626 | . . . . . . . 8 | |
77 | 75, 76 | mp3an2 1303 | . . . . . . 7 |
78 | 53, 73, 74, 77 | syl12anc 1214 | . . . . . 6 |
79 | 72, 78 | mpbid 146 | . . . . 5 |
80 | 51, 54, 67, 68, 70, 79 | leltaddd 8328 | . . . 4 |
81 | nncn 8728 | . . . . . . . 8 | |
82 | npcan1 8140 | . . . . . . . 8 | |
83 | 81, 82 | syl 14 | . . . . . . 7 |
84 | 83 | oveq1d 5789 | . . . . . 6 |
85 | 64 | recnd 7794 | . . . . . . 7 |
86 | ax-1cn 7713 | . . . . . . . 8 | |
87 | divdirap 8457 | . . . . . . . 8 # | |
88 | 86, 87 | mp3an2 1303 | . . . . . . 7 # |
89 | 85, 81, 65, 88 | syl12anc 1214 | . . . . . 6 |
90 | 81, 65 | dividapd 8546 | . . . . . 6 |
91 | 84, 89, 90 | 3eqtr3d 2180 | . . . . 5 |
92 | 91 | adantl 275 | . . . 4 |
93 | 80, 92 | breqtrd 3954 | . . 3 |
94 | 32 | flqcld 10050 | . . . 4 |
95 | qaddcl 9427 | . . . . 5 | |
96 | 36, 44, 95 | syl2anc 408 | . . . 4 |
97 | flqbi2 10064 | . . . 4 | |
98 | 94, 96, 97 | syl2anc 408 | . . 3 |
99 | 61, 93, 98 | mpbir2and 928 | . 2 |
100 | 49, 99 | eqtr2d 2173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 clt 7800 cle 7801 cmin 7933 # cap 8343 cdiv 8432 cn 8720 cz 9054 cq 9411 cfl 10041 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 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 |
This theorem depends on definitions: df-bi 116 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-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-id 4215 df-po 4218 df-iso 4219 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-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 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-n0 8978 df-z 9055 df-q 9412 df-rp 9442 df-fl 10043 |
This theorem is referenced by: modqmulnn 10115 |
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