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Mirrors > Home > ILE Home > Th. List > modq0 | Unicode version |
Description: is zero iff is evenly divisible by . (Contributed by Jim Kingdon, 17-Oct-2021.) |
Ref | Expression |
---|---|
modq0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqval 10273 | . . . . 5 | |
2 | 1 | eqeq1d 2179 | . . . 4 |
3 | qcn 9586 | . . . . . 6 | |
4 | 3 | 3ad2ant1 1013 | . . . . 5 |
5 | qcn 9586 | . . . . . . 7 | |
6 | 5 | 3ad2ant2 1014 | . . . . . 6 |
7 | simp3 994 | . . . . . . . . . 10 | |
8 | 7 | gt0ne0d 8424 | . . . . . . . . 9 |
9 | qdivcl 9595 | . . . . . . . . 9 | |
10 | 8, 9 | syld3an3 1278 | . . . . . . . 8 |
11 | 10 | flqcld 10226 | . . . . . . 7 |
12 | 11 | zcnd 9328 | . . . . . 6 |
13 | 6, 12 | mulcld 7933 | . . . . 5 |
14 | 4, 13 | subeq0ad 8233 | . . . 4 |
15 | 2, 14 | bitrd 187 | . . 3 |
16 | qre 9577 | . . . . . . 7 | |
17 | 16 | 3ad2ant2 1014 | . . . . . 6 |
18 | 17, 7 | gt0ap0d 8541 | . . . . 5 # |
19 | 4, 12, 6, 18 | divmulap2d 8734 | . . . 4 |
20 | eqcom 2172 | . . . 4 | |
21 | 19, 20 | bitr3di 194 | . . 3 |
22 | 15, 21 | bitrd 187 | . 2 |
23 | flqidz 10235 | . . 3 | |
24 | 10, 23 | syl 14 | . 2 |
25 | 22, 24 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 973 wceq 1348 wcel 2141 wne 2340 class class class wbr 3987 cfv 5196 (class class class)co 5851 cc 7765 cr 7766 cc0 7767 cmul 7772 clt 7947 cmin 8083 cdiv 8582 cz 9205 cq 9571 cfl 10217 cmo 10271 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-n0 9129 df-z 9206 df-q 9572 df-rp 9604 df-fl 10219 df-mod 10272 |
This theorem is referenced by: mulqmod0 10279 negqmod0 10280 modqid0 10299 q2txmodxeq0 10333 addmodlteq 10347 dvdsval3 11746 |
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