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Theorem mod2xnegi 13102
Description: Version of mod2xi 13100 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 10012 . . . 4  |-  ( ( L  e.  NN0  /\  K  e.  NN )  ->  ( L  +  K
)  e.  NN )
52, 3, 4mp2an 653 . . 3  |-  ( L  +  K )  e.  NN
61, 5eqeltrri 2367 . 2  |-  N  e.  NN
7 mod2xnegi.1 . 2  |-  A  e.  NN
8 mod2xnegi.2 . 2  |-  B  e. 
NN0
96nnzi 10063 . . . 4  |-  N  e.  ZZ
10 mod2xnegi.3 . . . 4  |-  D  e.  ZZ
11 zaddcl 10075 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  +  D
)  e.  ZZ )
129, 10, 11mp2an 653 . . 3  |-  ( N  +  D )  e.  ZZ
133nnnn0i 9989 . . . . 5  |-  K  e. 
NN0
1413, 13nn0addcli 10017 . . . 4  |-  ( K  +  K )  e. 
NN0
1514nn0zi 10064 . . 3  |-  ( K  +  K )  e.  ZZ
16 zsubcl 10077 . . 3  |-  ( ( ( N  +  D
)  e.  ZZ  /\  ( K  +  K
)  e.  ZZ )  ->  ( ( N  +  D )  -  ( K  +  K
) )  e.  ZZ )
1712, 15, 16mp2an 653 . 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 9772 . . . . . 6  |-  N  e.  CC
22 zcn 10045 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  e.  CC )
2310, 22ax-mp 8 . . . . . 6  |-  D  e.  CC
2421, 23addcli 8857 . . . . 5  |-  ( N  +  D )  e.  CC
253nncni 9772 . . . . . 6  |-  K  e.  CC
2625, 25addcli 8857 . . . . 5  |-  ( K  +  K )  e.  CC
2724, 26, 21subdiri 9245 . . . 4  |-  ( ( ( N  +  D
)  -  ( K  +  K ) )  x.  N )  =  ( ( ( N  +  D )  x.  N )  -  (
( K  +  K
)  x.  N ) )
2827oveq1i 5884 . . 3  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( ( ( ( N  +  D )  x.  N )  -  ( ( K  +  K )  x.  N
) )  +  M
)
2924, 21mulcli 8858 . . . 4  |-  ( ( N  +  D )  x.  N )  e.  CC
3018nn0cni 9993 . . . 4  |-  M  e.  CC
3126, 21mulcli 8858 . . . 4  |-  ( ( K  +  K )  x.  N )  e.  CC
3229, 30, 31addsubi 9154 . . 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 5885 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( N  x.  N )  +  ( K  x.  K ) )
3521, 25, 25adddii 8863 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( N  x.  K )  +  ( N  x.  K ) )
3634, 35oveq12i 5886 . . . . 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 8864 . . . . . . . 8  |-  ( ( N  +  D )  x.  N )  =  ( ( N  x.  N )  +  ( D  x.  N ) )
3837oveq1i 5884 . . . . . . 7  |-  ( ( ( N  +  D
)  x.  N )  +  M )  =  ( ( ( N  x.  N )  +  ( D  x.  N
) )  +  M
)
3921, 21mulcli 8858 . . . . . . . 8  |-  ( N  x.  N )  e.  CC
4023, 21mulcli 8858 . . . . . . . 8  |-  ( D  x.  N )  e.  CC
4139, 40, 30addassi 8861 . . . . . . 7  |-  ( ( ( N  x.  N
)  +  ( D  x.  N ) )  +  M )  =  ( ( N  x.  N )  +  ( ( D  x.  N
)  +  M ) )
4238, 41eqtr2i 2317 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( ( N  +  D )  x.  N )  +  M
)
4321, 26mulcomi 8859 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( K  +  K )  x.  N
)
4442, 43oveq12i 5886 . . . . 5  |-  ( ( ( N  x.  N
)  +  ( ( D  x.  N )  +  M ) )  -  ( N  x.  ( K  +  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
4536, 44eqtr3i 2318 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
46 mulsub 9238 . . . . . 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 654 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( ( ( N  x.  N )  +  ( K  x.  K
) )  -  (
( N  x.  K
)  +  ( N  x.  K ) ) )
482nn0cni 9993 . . . . . . . 8  |-  L  e.  CC
4921, 25, 48subadd2i 9150 . . . . . . 7  |-  ( ( N  -  K )  =  L  <->  ( L  +  K )  =  N )
501, 49mpbir 200 . . . . . 6  |-  ( N  -  K )  =  L
5150, 50oveq12i 5886 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( L  x.  L
)
5247, 51eqtr3i 2318 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( L  x.  L
)
5345, 52eqtr3i 2318 . . 3  |-  ( ( ( ( N  +  D )  x.  N
)  +  M )  -  ( ( K  +  K )  x.  N ) )  =  ( L  x.  L
)
5428, 32, 533eqtr2i 2322 . 2  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( L  x.  L
)
556, 7, 8, 17, 2, 18, 19, 20, 54mod2xi 13100 1  |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751    + caddc 8756    x. cmul 8758    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040    mod cmo 10989   ^cexp 11120
This theorem is referenced by:  1259lem4  13148  2503lem2  13152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121
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