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Theorem mod2xnegi 13335
Description: Version of mod2xi 13333 with a negaive mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
Hypotheses
Ref Expression
mod2xnegi.1  |-  A  e.  NN
mod2xnegi.2  |-  B  e. 
NN0
mod2xnegi.3  |-  D  e.  ZZ
mod2xnegi.4  |-  K  e.  NN
mod2xnegi.5  |-  M  e. 
NN0
mod2xnegi.6  |-  L  e. 
NN0
mod2xnegi.10  |-  ( ( A ^ B )  mod  N )  =  ( L  mod  N
)
mod2xnegi.7  |-  ( 2  x.  B )  =  E
mod2xnegi.8  |-  ( L  +  K )  =  N
mod2xnegi.9  |-  ( ( D  x.  N )  +  M )  =  ( K  x.  K
)
Assertion
Ref Expression
mod2xnegi  |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N
)

Proof of Theorem mod2xnegi
StepHypRef Expression
1 mod2xnegi.8 . . 3  |-  ( L  +  K )  =  N
2 mod2xnegi.6 . . . 4  |-  L  e. 
NN0
3 mod2xnegi.4 . . . 4  |-  K  e.  NN
4 nn0nnaddcl 10185 . . . 4  |-  ( ( L  e.  NN0  /\  K  e.  NN )  ->  ( L  +  K
)  e.  NN )
52, 3, 4mp2an 654 . . 3  |-  ( L  +  K )  e.  NN
61, 5eqeltrri 2459 . 2  |-  N  e.  NN
7 mod2xnegi.1 . 2  |-  A  e.  NN
8 mod2xnegi.2 . 2  |-  B  e. 
NN0
96nnzi 10238 . . . 4  |-  N  e.  ZZ
10 mod2xnegi.3 . . . 4  |-  D  e.  ZZ
11 zaddcl 10250 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  +  D
)  e.  ZZ )
129, 10, 11mp2an 654 . . 3  |-  ( N  +  D )  e.  ZZ
133nnnn0i 10162 . . . . 5  |-  K  e. 
NN0
1413, 13nn0addcli 10190 . . . 4  |-  ( K  +  K )  e. 
NN0
1514nn0zi 10239 . . 3  |-  ( K  +  K )  e.  ZZ
16 zsubcl 10252 . . 3  |-  ( ( ( N  +  D
)  e.  ZZ  /\  ( K  +  K
)  e.  ZZ )  ->  ( ( N  +  D )  -  ( K  +  K
) )  e.  ZZ )
1712, 15, 16mp2an 654 . 2  |-  ( ( N  +  D )  -  ( K  +  K ) )  e.  ZZ
18 mod2xnegi.5 . 2  |-  M  e. 
NN0
19 mod2xnegi.10 . 2  |-  ( ( A ^ B )  mod  N )  =  ( L  mod  N
)
20 mod2xnegi.7 . 2  |-  ( 2  x.  B )  =  E
216nncni 9943 . . . . . 6  |-  N  e.  CC
22 zcn 10220 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  e.  CC )
2310, 22ax-mp 8 . . . . . 6  |-  D  e.  CC
2421, 23addcli 9028 . . . . 5  |-  ( N  +  D )  e.  CC
253nncni 9943 . . . . . 6  |-  K  e.  CC
2625, 25addcli 9028 . . . . 5  |-  ( K  +  K )  e.  CC
2724, 26, 21subdiri 9416 . . . 4  |-  ( ( ( N  +  D
)  -  ( K  +  K ) )  x.  N )  =  ( ( ( N  +  D )  x.  N )  -  (
( K  +  K
)  x.  N ) )
2827oveq1i 6031 . . 3  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( ( ( ( N  +  D )  x.  N )  -  ( ( K  +  K )  x.  N
) )  +  M
)
2924, 21mulcli 9029 . . . 4  |-  ( ( N  +  D )  x.  N )  e.  CC
3018nn0cni 10166 . . . 4  |-  M  e.  CC
3126, 21mulcli 9029 . . . 4  |-  ( ( K  +  K )  x.  N )  e.  CC
3229, 30, 31addsubi 9325 . . 3  |-  ( ( ( ( N  +  D )  x.  N
)  +  M )  -  ( ( K  +  K )  x.  N ) )  =  ( ( ( ( N  +  D )  x.  N )  -  ( ( K  +  K )  x.  N
) )  +  M
)
33 mod2xnegi.9 . . . . . . 7  |-  ( ( D  x.  N )  +  M )  =  ( K  x.  K
)
3433oveq2i 6032 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( N  x.  N )  +  ( K  x.  K ) )
3521, 25, 25adddii 9034 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( N  x.  K )  +  ( N  x.  K ) )
3634, 35oveq12i 6033 . . . . 5  |-  ( ( ( N  x.  N
)  +  ( ( D  x.  N )  +  M ) )  -  ( N  x.  ( K  +  K
) ) )  =  ( ( ( N  x.  N )  +  ( K  x.  K
) )  -  (
( N  x.  K
)  +  ( N  x.  K ) ) )
3721, 23, 21adddiri 9035 . . . . . . . 8  |-  ( ( N  +  D )  x.  N )  =  ( ( N  x.  N )  +  ( D  x.  N ) )
3837oveq1i 6031 . . . . . . 7  |-  ( ( ( N  +  D
)  x.  N )  +  M )  =  ( ( ( N  x.  N )  +  ( D  x.  N
) )  +  M
)
3921, 21mulcli 9029 . . . . . . . 8  |-  ( N  x.  N )  e.  CC
4023, 21mulcli 9029 . . . . . . . 8  |-  ( D  x.  N )  e.  CC
4139, 40, 30addassi 9032 . . . . . . 7  |-  ( ( ( N  x.  N
)  +  ( D  x.  N ) )  +  M )  =  ( ( N  x.  N )  +  ( ( D  x.  N
)  +  M ) )
4238, 41eqtr2i 2409 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( ( N  +  D )  x.  N )  +  M
)
4321, 26mulcomi 9030 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( K  +  K )  x.  N
)
4442, 43oveq12i 6033 . . . . 5  |-  ( ( ( N  x.  N
)  +  ( ( D  x.  N )  +  M ) )  -  ( N  x.  ( K  +  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
4536, 44eqtr3i 2410 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
46 mulsub 9409 . . . . . 6  |-  ( ( ( N  e.  CC  /\  K  e.  CC )  /\  ( N  e.  CC  /\  K  e.  CC ) )  -> 
( ( N  -  K )  x.  ( N  -  K )
)  =  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) ) )
4721, 25, 21, 25, 46mp4an 655 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( ( ( N  x.  N )  +  ( K  x.  K
) )  -  (
( N  x.  K
)  +  ( N  x.  K ) ) )
482nn0cni 10166 . . . . . . . 8  |-  L  e.  CC
4921, 25, 48subadd2i 9321 . . . . . . 7  |-  ( ( N  -  K )  =  L  <->  ( L  +  K )  =  N )
501, 49mpbir 201 . . . . . 6  |-  ( N  -  K )  =  L
5150, 50oveq12i 6033 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( L  x.  L
)
5247, 51eqtr3i 2410 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( L  x.  L
)
5345, 52eqtr3i 2410 . . 3  |-  ( ( ( ( N  +  D )  x.  N
)  +  M )  -  ( ( K  +  K )  x.  N ) )  =  ( L  x.  L
)
5428, 32, 533eqtr2i 2414 . 2  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( L  x.  L
)
556, 7, 8, 17, 2, 18, 19, 20, 54mod2xi 13333 1  |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717  (class class class)co 6021   CCcc 8922    + caddc 8927    x. cmul 8929    - cmin 9224   NNcn 9933   2c2 9982   NN0cn0 10154   ZZcz 10215    mod cmo 11178   ^cexp 11310
This theorem is referenced by:  1259lem4  13381  2503lem2  13385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311
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