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Mirrors > Home > ILE Home > Th. List > divalgmod | Unicode version |
Description: The result of the operator satisfies the requirements for the remainder in the division algorithm for a positive divisor (compare divalg2 11848 and divalgb 11847). This demonstration theorem justifies the use of to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.) |
Ref | Expression |
---|---|
divalgmod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9555 | . . . . . . . 8 | |
2 | 1 | adantr 274 | . . . . . . 7 |
3 | nnq 9562 | . . . . . . . 8 | |
4 | 3 | adantl 275 | . . . . . . 7 |
5 | simpr 109 | . . . . . . . 8 | |
6 | 5 | nngt0d 8892 | . . . . . . 7 |
7 | 2, 4, 6 | modqcld 10253 | . . . . . 6 |
8 | snidg 3599 | . . . . . 6 | |
9 | 7, 8 | syl 14 | . . . . 5 |
10 | eleq1 2227 | . . . . 5 | |
11 | 9, 10 | syl5ibrcom 156 | . . . 4 |
12 | elsni 3588 | . . . 4 | |
13 | 11, 12 | impbid1 141 | . . 3 |
14 | modqlt 10258 | . . . . . . . . 9 | |
15 | 2, 4, 6, 14 | syl3anc 1227 | . . . . . . . 8 |
16 | znq 9553 | . . . . . . . . . 10 | |
17 | 16 | flqcld 10202 | . . . . . . . . 9 |
18 | nnz 9201 | . . . . . . . . . 10 | |
19 | 18 | adantl 275 | . . . . . . . . 9 |
20 | zmodcl 10269 | . . . . . . . . . . 11 | |
21 | 20 | nn0zd 9302 | . . . . . . . . . 10 |
22 | zsubcl 9223 | . . . . . . . . . 10 | |
23 | 21, 22 | syldan 280 | . . . . . . . . 9 |
24 | nncn 8856 | . . . . . . . . . . . 12 | |
25 | 24 | adantl 275 | . . . . . . . . . . 11 |
26 | 17 | zcnd 9305 | . . . . . . . . . . 11 |
27 | 25, 26 | mulcomd 7911 | . . . . . . . . . 10 |
28 | modqval 10249 | . . . . . . . . . . . 12 | |
29 | 2, 4, 6, 28 | syl3anc 1227 | . . . . . . . . . . 11 |
30 | 20 | nn0cnd 9160 | . . . . . . . . . . . 12 |
31 | zmulcl 9235 | . . . . . . . . . . . . . 14 | |
32 | 18, 17, 31 | syl2an2 584 | . . . . . . . . . . . . 13 |
33 | 32 | zcnd 9305 | . . . . . . . . . . . 12 |
34 | zcn 9187 | . . . . . . . . . . . . 13 | |
35 | 34 | adantr 274 | . . . . . . . . . . . 12 |
36 | 30, 33, 35 | subexsub 8261 | . . . . . . . . . . 11 |
37 | 29, 36 | mpbid 146 | . . . . . . . . . 10 |
38 | 27, 37 | eqtr3d 2199 | . . . . . . . . 9 |
39 | dvds0lem 11727 | . . . . . . . . 9 | |
40 | 17, 19, 23, 38, 39 | syl31anc 1230 | . . . . . . . 8 |
41 | divalg2 11848 | . . . . . . . . 9 | |
42 | breq1 3979 | . . . . . . . . . . 11 | |
43 | oveq2 5844 | . . . . . . . . . . . 12 | |
44 | 43 | breq2d 3988 | . . . . . . . . . . 11 |
45 | 42, 44 | anbi12d 465 | . . . . . . . . . 10 |
46 | 45 | riota2 5814 | . . . . . . . . 9 |
47 | 20, 41, 46 | syl2anc 409 | . . . . . . . 8 |
48 | 15, 40, 47 | mpbi2and 932 | . . . . . . 7 |
49 | 48 | eqcomd 2170 | . . . . . 6 |
50 | 49 | sneqd 3583 | . . . . 5 |
51 | snriota 5821 | . . . . . 6 | |
52 | 41, 51 | syl 14 | . . . . 5 |
53 | 50, 52 | eqtr4d 2200 | . . . 4 |
54 | 53 | eleq2d 2234 | . . 3 |
55 | 13, 54 | bitrd 187 | . 2 |
56 | breq1 3979 | . . . 4 | |
57 | oveq2 5844 | . . . . 5 | |
58 | 57 | breq2d 3988 | . . . 4 |
59 | 56, 58 | anbi12d 465 | . . 3 |
60 | 59 | elrab 2877 | . 2 |
61 | 55, 60 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wreu 2444 crab 2446 csn 3570 class class class wbr 3976 cfv 5182 crio 5791 (class class class)co 5836 cc 7742 cc0 7744 cmul 7749 clt 7924 cmin 8060 cdiv 8559 cn 8848 cn0 9105 cz 9182 cq 9548 cfl 10193 cmo 10247 cdvds 11713 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 |
This theorem is referenced by: divalgmodcl 11850 |
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