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Mirrors > Home > ILE Home > Th. List > modqcyc | Unicode version |
Description: The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.) |
Ref | Expression |
---|---|
modqcyc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 519 | . . . . 5 | |
2 | zq 9556 | . . . . . . 7 | |
3 | 2 | ad2antlr 481 | . . . . . 6 |
4 | simprl 521 | . . . . . 6 | |
5 | qmulcl 9567 | . . . . . 6 | |
6 | 3, 4, 5 | syl2anc 409 | . . . . 5 |
7 | qaddcl 9565 | . . . . 5 | |
8 | 1, 6, 7 | syl2anc 409 | . . . 4 |
9 | simprr 522 | . . . 4 | |
10 | modqval 10250 | . . . 4 | |
11 | 8, 4, 9, 10 | syl3anc 1227 | . . 3 |
12 | qcn 9564 | . . . . . . . . . . 11 | |
13 | 1, 12 | syl 14 | . . . . . . . . . 10 |
14 | qcn 9564 | . . . . . . . . . . 11 | |
15 | 6, 14 | syl 14 | . . . . . . . . . 10 |
16 | qcn 9564 | . . . . . . . . . . 11 | |
17 | 4, 16 | syl 14 | . . . . . . . . . 10 |
18 | qre 9555 | . . . . . . . . . . . 12 | |
19 | 4, 18 | syl 14 | . . . . . . . . . . 11 |
20 | 19, 9 | gt0ap0d 8519 | . . . . . . . . . 10 # |
21 | 13, 15, 17, 20 | divdirapd 8717 | . . . . . . . . 9 |
22 | simplr 520 | . . . . . . . . . . . 12 | |
23 | 22 | zcnd 9306 | . . . . . . . . . . 11 |
24 | 23, 17, 20 | divcanap4d 8684 | . . . . . . . . . 10 |
25 | 24 | oveq2d 5853 | . . . . . . . . 9 |
26 | 21, 25 | eqtrd 2197 | . . . . . . . 8 |
27 | 26 | fveq2d 5485 | . . . . . . 7 |
28 | 9 | gt0ne0d 8402 | . . . . . . . . 9 |
29 | qdivcl 9573 | . . . . . . . . 9 | |
30 | 1, 4, 28, 29 | syl3anc 1227 | . . . . . . . 8 |
31 | flqaddz 10223 | . . . . . . . 8 | |
32 | 30, 22, 31 | syl2anc 409 | . . . . . . 7 |
33 | 27, 32 | eqtrd 2197 | . . . . . 6 |
34 | 33 | oveq2d 5853 | . . . . 5 |
35 | 30 | flqcld 10203 | . . . . . . 7 |
36 | 35 | zcnd 9306 | . . . . . 6 |
37 | 17, 36, 23 | adddid 7915 | . . . . 5 |
38 | 17, 23 | mulcomd 7912 | . . . . . 6 |
39 | 38 | oveq2d 5853 | . . . . 5 |
40 | 34, 37, 39 | 3eqtrd 2201 | . . . 4 |
41 | 40 | oveq2d 5853 | . . 3 |
42 | 17, 36 | mulcld 7911 | . . . 4 |
43 | 13, 42, 15 | pnpcan2d 8239 | . . 3 |
44 | 11, 41, 43 | 3eqtrd 2201 | . 2 |
45 | modqval 10250 | . . 3 | |
46 | 1, 4, 9, 45 | syl3anc 1227 | . 2 |
47 | 44, 46 | eqtr4d 2200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wne 2334 class class class wbr 3977 cfv 5183 (class class class)co 5837 cc 7743 cr 7744 cc0 7745 caddc 7748 cmul 7750 clt 7925 cmin 8061 cdiv 8560 cz 9183 cq 9549 cfl 10194 cmo 10248 |
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-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 ax-arch 7864 |
This theorem depends on definitions: df-bi 116 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 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-po 4269 df-iso 4270 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 df-inn 8850 df-n0 9107 df-z 9184 df-q 9550 df-rp 9582 df-fl 10196 df-mod 10249 |
This theorem is referenced by: modqcyc2 10286 mulqaddmodid 10290 qnegmod 10295 modsumfzodifsn 10322 |
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