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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 12526 | . 2 โข 1 โ โ0 | |
8 | 2 | nnnn0i 12518 | . 2 โข ๐ด โ โ0 |
9 | modxp1i.9 | . 2 โข ((๐ดโ๐ต) mod ๐) = (๐พ mod ๐) | |
10 | 2 | nncni 12260 | . . . 4 โข ๐ด โ โ |
11 | exp1 14072 | . . . 4 โข (๐ด โ โ โ (๐ดโ1) = ๐ด) | |
12 | 10, 11 | ax-mp 5 | . . 3 โข (๐ดโ1) = ๐ด |
13 | 12 | oveq1i 7436 | . 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 17044 | 1 โข ((๐ดโ๐ธ) mod ๐) = (๐ mod ๐) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 (class class class)co 7426 โcc 11144 1c1 11147 + caddc 11149 ยท cmul 11151 โcn 12250 โ0cn0 12510 โคcz 12596 mod cmo 13874 โcexp 14066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 |
This theorem is referenced by: 1259lem1 17107 1259lem4 17110 2503lem2 17114 4001lem1 17117 2exp340mod341 47102 nfermltl8rev 47111 |
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