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Mirrors > Home > MPE Home > Th. List > modxp1i | Structured version Visualization version GIF version |
Description: Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) |
Ref | Expression |
---|---|
modxai.1 | โข ๐ โ โ |
modxai.2 | โข ๐ด โ โ |
modxai.3 | โข ๐ต โ โ0 |
modxai.4 | โข ๐ท โ โค |
modxai.5 | โข ๐พ โ โ0 |
modxai.6 | โข ๐ โ โ0 |
modxp1i.9 | โข ((๐ดโ๐ต) mod ๐) = (๐พ mod ๐) |
modxp1i.7 | โข (๐ต + 1) = ๐ธ |
modxp1i.8 | โข ((๐ท ยท ๐) + ๐) = (๐พ ยท ๐ด) |
Ref | Expression |
---|---|
modxp1i | โข ((๐ดโ๐ธ) mod ๐) = (๐ mod ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modxai.1 | . 2 โข ๐ โ โ | |
2 | modxai.2 | . 2 โข ๐ด โ โ | |
3 | modxai.3 | . 2 โข ๐ต โ โ0 | |
4 | modxai.4 | . 2 โข ๐ท โ โค | |
5 | modxai.5 | . 2 โข ๐พ โ โ0 | |
6 | modxai.6 | . 2 โข ๐ โ โ0 | |
7 | 1nn0 12434 | . 2 โข 1 โ โ0 | |
8 | 2 | nnnn0i 12426 | . 2 โข ๐ด โ โ0 |
9 | modxp1i.9 | . 2 โข ((๐ดโ๐ต) mod ๐) = (๐พ mod ๐) | |
10 | 2 | nncni 12168 | . . . 4 โข ๐ด โ โ |
11 | exp1 13979 | . . . 4 โข (๐ด โ โ โ (๐ดโ1) = ๐ด) | |
12 | 10, 11 | ax-mp 5 | . . 3 โข (๐ดโ1) = ๐ด |
13 | 12 | oveq1i 7368 | . 2 โข ((๐ดโ1) mod ๐) = (๐ด mod ๐) |
14 | modxp1i.7 | . 2 โข (๐ต + 1) = ๐ธ | |
15 | modxp1i.8 | . 2 โข ((๐ท ยท ๐) + ๐) = (๐พ ยท ๐ด) | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15 | modxai 16945 | 1 โข ((๐ดโ๐ธ) mod ๐) = (๐ mod ๐) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7358 โcc 11054 1c1 11057 + caddc 11059 ยท cmul 11061 โcn 12158 โ0cn0 12418 โคcz 12504 mod cmo 13780 โcexp 13973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fl 13703 df-mod 13781 df-seq 13913 df-exp 13974 |
This theorem is referenced by: 1259lem1 17008 1259lem4 17011 2503lem2 17015 4001lem1 17018 2exp340mod341 46011 nfermltl8rev 46020 |
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