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Mirrors > Home > ILE Home > Th. List > q2submod | Unicode version |
Description: If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.) |
Ref | Expression |
---|---|
q2submod |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qcn 9630 |
. . . . . . 7
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2 | 1 | 3ad2ant2 1019 |
. . . . . 6
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3 | 2 | adantr 276 |
. . . . 5
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4 | 3 | mulridd 7971 |
. . . 4
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5 | 4 | oveq2d 5888 |
. . 3
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6 | 5 | oveq1d 5887 |
. 2
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7 | simpl1 1000 |
. . 3
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8 | 1zzd 9276 |
. . 3
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9 | simpl2 1001 |
. . 3
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10 | simpl3 1002 |
. . 3
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11 | modqcyc2 10355 |
. . 3
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12 | 7, 8, 9, 10, 11 | syl22anc 1239 |
. 2
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13 | qsubcl 9634 |
. . . 4
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14 | 7, 9, 13 | syl2anc 411 |
. . 3
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15 | simpr 110 |
. . . 4
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16 | qre 9621 |
. . . . . . . 8
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17 | 7, 16 | syl 14 |
. . . . . . 7
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18 | qre 9621 |
. . . . . . . 8
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19 | 9, 18 | syl 14 |
. . . . . . 7
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20 | 17, 19 | subge0d 8488 |
. . . . . 6
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21 | 20 | bicomd 141 |
. . . . 5
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22 | 3 | 2timesd 9157 |
. . . . . . 7
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23 | 22 | breq2d 4014 |
. . . . . 6
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24 | 17, 19, 19 | ltsubaddd 8494 |
. . . . . 6
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25 | 23, 24 | bitr4d 191 |
. . . . 5
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26 | 21, 25 | anbi12d 473 |
. . . 4
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27 | 15, 26 | mpbid 147 |
. . 3
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28 | modqid 10344 |
. . 3
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29 | 14, 9, 27, 28 | syl21anc 1237 |
. 2
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30 | 6, 12, 29 | 3eqtr3d 2218 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 ax-arch 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-po 4295 df-iso 4296 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-inn 8916 df-2 8974 df-n0 9173 df-z 9250 df-q 9616 df-rp 9650 df-fl 10265 df-mod 10318 |
This theorem is referenced by: modifeq2int 10381 modaddmodup 10382 |
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