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Theorem mod2xnegi 13086
Description: Version of mod2xi 13084 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 9996 . . . 4  |-  ( ( L  e.  NN0  /\  K  e.  NN )  ->  ( L  +  K
)  e.  NN )
52, 3, 4mp2an 653 . . 3  |-  ( L  +  K )  e.  NN
61, 5eqeltrri 2354 . 2  |-  N  e.  NN
7 mod2xnegi.1 . 2  |-  A  e.  NN
8 mod2xnegi.2 . 2  |-  B  e. 
NN0
96nnzi 10047 . . . 4  |-  N  e.  ZZ
10 mod2xnegi.3 . . . 4  |-  D  e.  ZZ
11 zaddcl 10059 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ )  ->  ( N  +  D
)  e.  ZZ )
129, 10, 11mp2an 653 . . 3  |-  ( N  +  D )  e.  ZZ
133nnnn0i 9973 . . . . 5  |-  K  e. 
NN0
1413, 13nn0addcli 10001 . . . 4  |-  ( K  +  K )  e. 
NN0
1514nn0zi 10048 . . 3  |-  ( K  +  K )  e.  ZZ
16 zsubcl 10061 . . 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 9756 . . . . . 6  |-  N  e.  CC
22 zcn 10029 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  e.  CC )
2310, 22ax-mp 8 . . . . . 6  |-  D  e.  CC
2421, 23addcli 8841 . . . . 5  |-  ( N  +  D )  e.  CC
253nncni 9756 . . . . . 6  |-  K  e.  CC
2625, 25addcli 8841 . . . . 5  |-  ( K  +  K )  e.  CC
2724, 26, 21subdiri 9229 . . . 4  |-  ( ( ( N  +  D
)  -  ( K  +  K ) )  x.  N )  =  ( ( ( N  +  D )  x.  N )  -  (
( K  +  K
)  x.  N ) )
2827oveq1i 5868 . . 3  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( ( ( ( N  +  D )  x.  N )  -  ( ( K  +  K )  x.  N
) )  +  M
)
2924, 21mulcli 8842 . . . 4  |-  ( ( N  +  D )  x.  N )  e.  CC
3018nn0cni 9977 . . . 4  |-  M  e.  CC
3126, 21mulcli 8842 . . . 4  |-  ( ( K  +  K )  x.  N )  e.  CC
3229, 30, 31addsubi 9138 . . 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 5869 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( N  x.  N )  +  ( K  x.  K ) )
3521, 25, 25adddii 8847 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( N  x.  K )  +  ( N  x.  K ) )
3634, 35oveq12i 5870 . . . . 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 8848 . . . . . . . 8  |-  ( ( N  +  D )  x.  N )  =  ( ( N  x.  N )  +  ( D  x.  N ) )
3837oveq1i 5868 . . . . . . 7  |-  ( ( ( N  +  D
)  x.  N )  +  M )  =  ( ( ( N  x.  N )  +  ( D  x.  N
) )  +  M
)
3921, 21mulcli 8842 . . . . . . . 8  |-  ( N  x.  N )  e.  CC
4023, 21mulcli 8842 . . . . . . . 8  |-  ( D  x.  N )  e.  CC
4139, 40, 30addassi 8845 . . . . . . 7  |-  ( ( ( N  x.  N
)  +  ( D  x.  N ) )  +  M )  =  ( ( N  x.  N )  +  ( ( D  x.  N
)  +  M ) )
4238, 41eqtr2i 2304 . . . . . 6  |-  ( ( N  x.  N )  +  ( ( D  x.  N )  +  M ) )  =  ( ( ( N  +  D )  x.  N )  +  M
)
4321, 26mulcomi 8843 . . . . . 6  |-  ( N  x.  ( K  +  K ) )  =  ( ( K  +  K )  x.  N
)
4442, 43oveq12i 5870 . . . . 5  |-  ( ( ( N  x.  N
)  +  ( ( D  x.  N )  +  M ) )  -  ( N  x.  ( K  +  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
4536, 44eqtr3i 2305 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( ( ( ( N  +  D )  x.  N )  +  M )  -  (
( K  +  K
)  x.  N ) )
46 mulsub 9222 . . . . . 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 9977 . . . . . . . 8  |-  L  e.  CC
4921, 25, 48subadd2i 9134 . . . . . . 7  |-  ( ( N  -  K )  =  L  <->  ( L  +  K )  =  N )
501, 49mpbir 200 . . . . . 6  |-  ( N  -  K )  =  L
5150, 50oveq12i 5870 . . . . 5  |-  ( ( N  -  K )  x.  ( N  -  K ) )  =  ( L  x.  L
)
5247, 51eqtr3i 2305 . . . 4  |-  ( ( ( N  x.  N
)  +  ( K  x.  K ) )  -  ( ( N  x.  K )  +  ( N  x.  K
) ) )  =  ( L  x.  L
)
5345, 52eqtr3i 2305 . . 3  |-  ( ( ( ( N  +  D )  x.  N
)  +  M )  -  ( ( K  +  K )  x.  N ) )  =  ( L  x.  L
)
5428, 32, 533eqtr2i 2309 . 2  |-  ( ( ( ( N  +  D )  -  ( K  +  K )
)  x.  N )  +  M )  =  ( L  x.  L
)
556, 7, 8, 17, 2, 18, 19, 20, 54mod2xi 13084 1  |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024    mod cmo 10973   ^cexp 11104
This theorem is referenced by:  1259lem4  13132  2503lem2  13136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105
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