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Mirrors > Home > ILE Home > Th. List > m1modge3gt1 | Unicode version |
Description: Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.) |
Ref | Expression |
---|---|
m1modge3gt1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1p1e2 9012 |
. . . 4
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2 | 2p1e3 9028 |
. . . . . 6
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3 | eluzle 9516 |
. . . . . 6
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4 | 2, 3 | eqbrtrid 4035 |
. . . . 5
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5 | 2z 9257 |
. . . . . 6
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6 | eluzelz 9513 |
. . . . . 6
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7 | zltp1le 9283 |
. . . . . 6
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8 | 5, 6, 7 | sylancr 414 |
. . . . 5
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9 | 4, 8 | mpbird 167 |
. . . 4
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10 | 1, 9 | eqbrtrid 4035 |
. . 3
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11 | 1red 7950 |
. . . 4
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12 | eluzelre 9514 |
. . . 4
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13 | 11, 11, 12 | ltaddsub2d 8480 |
. . 3
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14 | 10, 13 | mpbid 147 |
. 2
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15 | eluzge3nn 9548 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | m1modnnsub1 10343 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | syl 14 |
. 2
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18 | 14, 17 | breqtrrd 4028 |
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 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-3 8955 df-n0 9153 df-z 9230 df-uz 9505 df-q 9596 df-rp 9628 df-fl 10243 df-mod 10296 |
This theorem is referenced by: (None) |
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