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Mirrors > Home > ILE Home > Th. List > modqge0 | Unicode version |
Description: The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
modqge0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 984 | . . . . . . 7 | |
2 | 1 | gt0ne0d 8382 | . . . . . 6 |
3 | qdivcl 9547 | . . . . . 6 | |
4 | 2, 3 | syld3an3 1265 | . . . . 5 |
5 | flqle 10172 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 4 | flqcld 10171 | . . . . . 6 |
8 | 7 | zred 9281 | . . . . 5 |
9 | qre 9529 | . . . . . 6 | |
10 | 9 | 3ad2ant1 1003 | . . . . 5 |
11 | qre 9529 | . . . . . 6 | |
12 | 11 | 3ad2ant2 1004 | . . . . 5 |
13 | lemuldiv2 8748 | . . . . 5 | |
14 | 8, 10, 12, 1, 13 | syl112anc 1224 | . . . 4 |
15 | 6, 14 | mpbird 166 | . . 3 |
16 | 12, 8 | remulcld 7903 | . . . 4 |
17 | 10, 16 | subge0d 8405 | . . 3 |
18 | 15, 17 | mpbird 166 | . 2 |
19 | modqval 10218 | . 2 | |
20 | 18, 19 | breqtrrd 3992 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 963 wcel 2128 wne 2327 class class class wbr 3965 cfv 5169 (class class class)co 5821 cr 7726 cc0 7727 cmul 7732 clt 7907 cle 7908 cmin 8041 cdiv 8540 cq 9523 cfl 10162 cmo 10216 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-mulrcl 7826 ax-addcom 7827 ax-mulcom 7828 ax-addass 7829 ax-mulass 7830 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-1rid 7834 ax-0id 7835 ax-rnegex 7836 ax-precex 7837 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 ax-pre-mulgt0 7844 ax-pre-mulext 7845 ax-arch 7846 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-po 4256 df-iso 4257 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-reap 8445 df-ap 8452 df-div 8541 df-inn 8829 df-n0 9086 df-z 9163 df-q 9524 df-rp 9556 df-fl 10164 df-mod 10217 |
This theorem is referenced by: modqelico 10228 zmodcl 10238 modqid2 10245 modqabs 10251 modqmuladdim 10261 modqltm1p1mod 10270 modqsubdir 10287 modqeqmodmin 10288 |
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