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Theorem mod2xnegi 13399
Description: Version of mod2xi 13397 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 10244 . . . 4  |-  ( ( L  e.  NN0  /\  K  e.  NN )  ->  ( L  +  K
)  e.  NN )
52, 3, 4mp2an 654 . . 3  |-  ( L  +  K )  e.  NN
61, 5eqeltrri 2506 . 2  |-  N  e.  NN
7 mod2xnegi.1 . 2  |-  A  e.  NN
8 mod2xnegi.2 . 2  |-  B  e. 
NN0
96nnzi 10297 . . . 4  |-  N  e.  ZZ
10 mod2xnegi.3 . . . 4  |-  D  e.  ZZ
11 zaddcl 10309 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  +  D
)  e.  ZZ )
129, 10, 11mp2an 654 . . 3  |-  ( N  +  D )  e.  ZZ
133nnnn0i 10221 . . . . 5  |-  K  e. 
NN0
1413, 13nn0addcli 10249 . . . 4  |-  ( K  +  K )  e. 
NN0
1514nn0zi 10298 . . 3  |-  ( K  +  K )  e.  ZZ
16 zsubcl 10311 . . 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 10002 . . . . . 6  |-  N  e.  CC
22 zcn 10279 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  e.  CC )
2310, 22ax-mp 8 . . . . . 6  |-  D  e.  CC
2421, 23addcli 9086 . . . . 5  |-  ( N  +  D )  e.  CC
253nncni 10002 . . . . . 6  |-  K  e.  CC
2625, 25addcli 9086 . . . . 5  |-  ( K  +  K )  e.  CC
2724, 26, 21subdiri 9475 . . . 4  |-  ( ( ( N  +  D
)  -  ( K  +  K ) )  x.  N )  =  ( ( ( N  +  D )  x.  N )  -  (
( K  +  K
)  x.  N ) )
2827oveq1i 6083 . . 3  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( ( ( ( N  +  D )  x.  N )  -  ( ( K  +  K )  x.  N
) )  +  M
)
2924, 21mulcli 9087 . . . 4  |-  ( ( N  +  D )  x.  N )  e.  CC
3018nn0cni 10225 . . . 4  |-  M  e.  CC
3126, 21mulcli 9087 . . . 4  |-  ( ( K  +  K )  x.  N )  e.  CC
3229, 30, 31addsubi 9384 . . 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 6084 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( N  x.  N )  +  ( K  x.  K ) )
3521, 25, 25adddii 9092 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( N  x.  K )  +  ( N  x.  K ) )
3634, 35oveq12i 6085 . . . . 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 9093 . . . . . . . 8  |-  ( ( N  +  D )  x.  N )  =  ( ( N  x.  N )  +  ( D  x.  N ) )
3837oveq1i 6083 . . . . . . 7  |-  ( ( ( N  +  D
)  x.  N )  +  M )  =  ( ( ( N  x.  N )  +  ( D  x.  N
) )  +  M
)
3921, 21mulcli 9087 . . . . . . . 8  |-  ( N  x.  N )  e.  CC
4023, 21mulcli 9087 . . . . . . . 8  |-  ( D  x.  N )  e.  CC
4139, 40, 30addassi 9090 . . . . . . 7  |-  ( ( ( N  x.  N
)  +  ( D  x.  N ) )  +  M )  =  ( ( N  x.  N )  +  ( ( D  x.  N
)  +  M ) )
4238, 41eqtr2i 2456 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( ( N  +  D )  x.  N )  +  M
)
4321, 26mulcomi 9088 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( K  +  K )  x.  N
)
4442, 43oveq12i 6085 . . . . 5  |-  ( ( ( N  x.  N
)  +  ( ( D  x.  N )  +  M ) )  -  ( N  x.  ( K  +  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
4536, 44eqtr3i 2457 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
46 mulsub 9468 . . . . . 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 10225 . . . . . . . 8  |-  L  e.  CC
4921, 25, 48subadd2i 9380 . . . . . . 7  |-  ( ( N  -  K )  =  L  <->  ( L  +  K )  =  N )
501, 49mpbir 201 . . . . . 6  |-  ( N  -  K )  =  L
5150, 50oveq12i 6085 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( L  x.  L
)
5247, 51eqtr3i 2457 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( L  x.  L
)
5345, 52eqtr3i 2457 . . 3  |-  ( ( ( ( N  +  D )  x.  N
)  +  M )  -  ( ( K  +  K )  x.  N ) )  =  ( L  x.  L
)
5428, 32, 533eqtr2i 2461 . 2  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( L  x.  L
)
556, 7, 8, 17, 2, 18, 19, 20, 54mod2xi 13397 1  |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725  (class class class)co 6073   CCcc 8980    + caddc 8985    x. cmul 8987    - cmin 9283   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274    mod cmo 11242   ^cexp 11374
This theorem is referenced by:  1259lem4  13445  2503lem2  13449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375
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