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Theorem mod2xnegi 14415
Description: Version of mod2xi 14413 with a negative 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 10826 . . . 4  |-  ( ( L  e.  NN0  /\  K  e.  NN )  ->  ( L  +  K
)  e.  NN )
52, 3, 4mp2an 672 . . 3  |-  ( L  +  K )  e.  NN
61, 5eqeltrri 2552 . 2  |-  N  e.  NN
7 mod2xnegi.1 . 2  |-  A  e.  NN
8 mod2xnegi.2 . 2  |-  B  e. 
NN0
96nnzi 10887 . . . 4  |-  N  e.  ZZ
10 mod2xnegi.3 . . . 4  |-  D  e.  ZZ
11 zaddcl 10902 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  +  D
)  e.  ZZ )
129, 10, 11mp2an 672 . . 3  |-  ( N  +  D )  e.  ZZ
133nnnn0i 10802 . . . . 5  |-  K  e. 
NN0
1413, 13nn0addcli 10832 . . . 4  |-  ( K  +  K )  e. 
NN0
1514nn0zi 10888 . . 3  |-  ( K  +  K )  e.  ZZ
16 zsubcl 10904 . . 3  |-  ( ( ( N  +  D
)  e.  ZZ  /\  ( K  +  K
)  e.  ZZ )  ->  ( ( N  +  D )  -  ( K  +  K
) )  e.  ZZ )
1712, 15, 16mp2an 672 . 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 10545 . . . . . 6  |-  N  e.  CC
22 zcn 10868 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  e.  CC )
2310, 22ax-mp 5 . . . . . 6  |-  D  e.  CC
2421, 23addcli 9599 . . . . 5  |-  ( N  +  D )  e.  CC
253nncni 10545 . . . . . 6  |-  K  e.  CC
2625, 25addcli 9599 . . . . 5  |-  ( K  +  K )  e.  CC
2724, 26, 21subdiri 10005 . . . 4  |-  ( ( ( N  +  D
)  -  ( K  +  K ) )  x.  N )  =  ( ( ( N  +  D )  x.  N )  -  (
( K  +  K
)  x.  N ) )
2827oveq1i 6293 . . 3  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( ( ( ( N  +  D )  x.  N )  -  ( ( K  +  K )  x.  N
) )  +  M
)
2924, 21mulcli 9600 . . . 4  |-  ( ( N  +  D )  x.  N )  e.  CC
3018nn0cni 10806 . . . 4  |-  M  e.  CC
3126, 21mulcli 9600 . . . 4  |-  ( ( K  +  K )  x.  N )  e.  CC
3229, 30, 31addsubi 9910 . . 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 6294 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( N  x.  N )  +  ( K  x.  K ) )
3521, 25, 25adddii 9605 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( N  x.  K )  +  ( N  x.  K ) )
3634, 35oveq12i 6295 . . . . 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 9606 . . . . . . . 8  |-  ( ( N  +  D )  x.  N )  =  ( ( N  x.  N )  +  ( D  x.  N ) )
3837oveq1i 6293 . . . . . . 7  |-  ( ( ( N  +  D
)  x.  N )  +  M )  =  ( ( ( N  x.  N )  +  ( D  x.  N
) )  +  M
)
3921, 21mulcli 9600 . . . . . . . 8  |-  ( N  x.  N )  e.  CC
4023, 21mulcli 9600 . . . . . . . 8  |-  ( D  x.  N )  e.  CC
4139, 40, 30addassi 9603 . . . . . . 7  |-  ( ( ( N  x.  N
)  +  ( D  x.  N ) )  +  M )  =  ( ( N  x.  N )  +  ( ( D  x.  N
)  +  M ) )
4238, 41eqtr2i 2497 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( ( N  +  D )  x.  N )  +  M
)
4321, 26mulcomi 9601 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( K  +  K )  x.  N
)
4442, 43oveq12i 6295 . . . . 5  |-  ( ( ( N  x.  N
)  +  ( ( D  x.  N )  +  M ) )  -  ( N  x.  ( K  +  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
4536, 44eqtr3i 2498 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
46 mulsub 9998 . . . . . 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 673 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( ( ( N  x.  N )  +  ( K  x.  K
) )  -  (
( N  x.  K
)  +  ( N  x.  K ) ) )
482nn0cni 10806 . . . . . . . 8  |-  L  e.  CC
4921, 25, 48subadd2i 9906 . . . . . . 7  |-  ( ( N  -  K )  =  L  <->  ( L  +  K )  =  N )
501, 49mpbir 209 . . . . . 6  |-  ( N  -  K )  =  L
5150, 50oveq12i 6295 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( L  x.  L
)
5247, 51eqtr3i 2498 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( L  x.  L
)
5345, 52eqtr3i 2498 . . 3  |-  ( ( ( ( N  +  D )  x.  N
)  +  M )  -  ( ( K  +  K )  x.  N ) )  =  ( L  x.  L
)
5428, 32, 533eqtr2i 2502 . 2  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( L  x.  L
)
556, 7, 8, 17, 2, 18, 19, 20, 54mod2xi 14413 1  |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767  (class class class)co 6283   CCcc 9489    + caddc 9494    x. cmul 9496    - cmin 9804   NNcn 10535   2c2 10584   NN0cn0 10794   ZZcz 10863    mod cmo 11963   ^cexp 12133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134
This theorem is referenced by:  1259lem4  14473  2503lem2  14477
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