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Theorem gausslemma2dlem7 15276
Description: Lemma 7 for gausslemma2d 15277. (Contributed by AV, 13-Jul-2021.)
Hypotheses
Ref Expression
gausslemma2d.p  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
gausslemma2d.h  |-  H  =  ( ( P  - 
1 )  /  2
)
gausslemma2d.r  |-  R  =  ( x  e.  ( 1 ... H ) 
|->  if ( ( x  x.  2 )  < 
( P  /  2
) ,  ( x  x.  2 ) ,  ( P  -  (
x  x.  2 ) ) ) )
gausslemma2d.m  |-  M  =  ( |_ `  ( P  /  4 ) )
gausslemma2d.n  |-  N  =  ( H  -  M
)
Assertion
Ref Expression
gausslemma2dlem7  |-  ( ph  ->  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P )  =  1 )
Distinct variable groups:    x, H    x, P    ph, x    x, M
Allowed substitution hints:    R( x)    N( x)

Proof of Theorem gausslemma2dlem7
StepHypRef Expression
1 gausslemma2d.p . . 3  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
2 gausslemma2d.h . . 3  |-  H  =  ( ( P  - 
1 )  /  2
)
3 gausslemma2d.r . . 3  |-  R  =  ( x  e.  ( 1 ... H ) 
|->  if ( ( x  x.  2 )  < 
( P  /  2
) ,  ( x  x.  2 ) ,  ( P  -  (
x  x.  2 ) ) ) )
4 gausslemma2d.m . . 3  |-  M  =  ( |_ `  ( P  /  4 ) )
5 gausslemma2d.n . . 3  |-  N  =  ( H  -  M
)
61, 2, 3, 4, 5gausslemma2dlem6 15275 . 2  |-  ( ph  ->  ( ( ! `  H )  mod  P
)  =  ( ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  x.  ( ! `
 H ) )  mod  P ) )
71, 2gausslemma2dlem0b 15258 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  NN )
87nnnn0d 9299 . . . . . . . . . 10  |-  ( ph  ->  H  e.  NN0 )
98faccld 10813 . . . . . . . . 9  |-  ( ph  ->  ( ! `  H
)  e.  NN )
109nncnd 9001 . . . . . . . 8  |-  ( ph  ->  ( ! `  H
)  e.  CC )
1110mullidd 8042 . . . . . . 7  |-  ( ph  ->  ( 1  x.  ( ! `  H )
)  =  ( ! `
 H ) )
1211eqcomd 2202 . . . . . 6  |-  ( ph  ->  ( ! `  H
)  =  ( 1  x.  ( ! `  H ) ) )
1312oveq1d 5937 . . . . 5  |-  ( ph  ->  ( ( ! `  H )  mod  P
)  =  ( ( 1  x.  ( ! `
 H ) )  mod  P ) )
1413eqeq1d 2205 . . . 4  |-  ( ph  ->  ( ( ( ! `
 H )  mod 
P )  =  ( ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( ! `  H
) )  mod  P
)  <->  ( ( 1  x.  ( ! `  H ) )  mod 
P )  =  ( ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( ! `  H
) )  mod  P
) ) )
15 1zzd 9350 . . . . 5  |-  ( ph  ->  1  e.  ZZ )
16 neg1z 9355 . . . . . . 7  |-  -u 1  e.  ZZ
171, 4, 2, 5gausslemma2dlem0h 15264 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
18 zexpcl 10631 . . . . . . 7  |-  ( (
-u 1  e.  ZZ  /\  N  e.  NN0 )  ->  ( -u 1 ^ N )  e.  ZZ )
1916, 17, 18sylancr 414 . . . . . 6  |-  ( ph  ->  ( -u 1 ^ N )  e.  ZZ )
20 2z 9351 . . . . . . 7  |-  2  e.  ZZ
21 zexpcl 10631 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  H  e.  NN0 )  -> 
( 2 ^ H
)  e.  ZZ )
2220, 8, 21sylancr 414 . . . . . 6  |-  ( ph  ->  ( 2 ^ H
)  e.  ZZ )
2319, 22zmulcld 9451 . . . . 5  |-  ( ph  ->  ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  e.  ZZ )
249nnzd 9444 . . . . 5  |-  ( ph  ->  ( ! `  H
)  e.  ZZ )
251gausslemma2dlem0a 15257 . . . . 5  |-  ( ph  ->  P  e.  NN )
261, 2gausslemma2dlem0c 15259 . . . . 5  |-  ( ph  ->  ( ( ! `  H )  gcd  P
)  =  1 )
27 cncongrcoprm 12250 . . . . 5  |-  ( ( ( 1  e.  ZZ  /\  ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  e.  ZZ  /\  ( ! `  H )  e.  ZZ )  /\  ( P  e.  NN  /\  ( ( ! `  H )  gcd  P
)  =  1 ) )  ->  ( (
( 1  x.  ( ! `  H )
)  mod  P )  =  ( ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  x.  ( ! `
 H ) )  mod  P )  <->  ( 1  mod  P )  =  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P ) ) )
2815, 23, 24, 25, 26, 27syl32anc 1257 . . . 4  |-  ( ph  ->  ( ( ( 1  x.  ( ! `  H ) )  mod 
P )  =  ( ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( ! `  H
) )  mod  P
)  <->  ( 1  mod 
P )  =  ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  mod  P ) ) )
2914, 28bitrd 188 . . 3  |-  ( ph  ->  ( ( ( ! `
 H )  mod 
P )  =  ( ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( ! `  H
) )  mod  P
)  <->  ( 1  mod 
P )  =  ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  mod  P ) ) )
30 simpr 110 . . . . 5  |-  ( (
ph  /\  ( 1  mod  P )  =  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P ) )  -> 
( 1  mod  P
)  =  ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  mod  P ) )
31 nnq 9704 . . . . . . . 8  |-  ( P  e.  NN  ->  P  e.  QQ )
3225, 31syl 14 . . . . . . 7  |-  ( ph  ->  P  e.  QQ )
331eldifad 3168 . . . . . . . 8  |-  ( ph  ->  P  e.  Prime )
34 prmgt1 12276 . . . . . . . 8  |-  ( P  e.  Prime  ->  1  < 
P )
3533, 34syl 14 . . . . . . 7  |-  ( ph  ->  1  <  P )
36 q1mod 10433 . . . . . . 7  |-  ( ( P  e.  QQ  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
3732, 35, 36syl2anc 411 . . . . . 6  |-  ( ph  ->  ( 1  mod  P
)  =  1 )
3837adantr 276 . . . . 5  |-  ( (
ph  /\  ( 1  mod  P )  =  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P ) )  -> 
( 1  mod  P
)  =  1 )
3930, 38eqtr3d 2231 . . . 4  |-  ( (
ph  /\  ( 1  mod  P )  =  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P ) )  -> 
( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P )  =  1 )
4039ex 115 . . 3  |-  ( ph  ->  ( ( 1  mod 
P )  =  ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  mod  P )  ->  ( ( (
-u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  1 ) )
4129, 40sylbid 150 . 2  |-  ( ph  ->  ( ( ( ! `
 H )  mod 
P )  =  ( ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( ! `  H
) )  mod  P
)  ->  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  1 ) )
426, 41mpd 13 1  |-  ( ph  ->  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2167    \ cdif 3154   ifcif 3561   {csn 3622   class class class wbr 4033    |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   1c1 7878    x. cmul 7882    < clt 8059    - cmin 8195   -ucneg 8196    / cdiv 8696   NNcn 8987   2c2 9038   4c4 9040   NN0cn0 9246   ZZcz 9323   QQcq 9690   ...cfz 10080   |_cfl 10343    mod cmo 10399   ^cexp 10615   !cfa 10802    gcd cgcd 12085   Primecprime 12251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-mulrcl 7976  ax-addcom 7977  ax-mulcom 7978  ax-addass 7979  ax-mulass 7980  ax-distr 7981  ax-i2m1 7982  ax-0lt1 7983  ax-1rid 7984  ax-0id 7985  ax-rnegex 7986  ax-precex 7987  ax-cnre 7988  ax-pre-ltirr 7989  ax-pre-ltwlin 7990  ax-pre-lttrn 7991  ax-pre-apti 7992  ax-pre-ltadd 7993  ax-pre-mulgt0 7994  ax-pre-mulext 7995  ax-arch 7996  ax-caucvg 7997
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7048  df-pnf 8061  df-mnf 8062  df-xr 8063  df-ltxr 8064  df-le 8065  df-sub 8197  df-neg 8198  df-reap 8599  df-ap 8606  df-div 8697  df-inn 8988  df-2 9046  df-3 9047  df-4 9048  df-5 9049  df-6 9050  df-n0 9247  df-z 9324  df-uz 9599  df-q 9691  df-rp 9726  df-ioo 9964  df-fz 10081  df-fzo 10215  df-fl 10345  df-mod 10400  df-seqfrec 10525  df-exp 10616  df-fac 10803  df-ihash 10853  df-cj 10992  df-re 10993  df-im 10994  df-rsqrt 11148  df-abs 11149  df-clim 11428  df-proddc 11700  df-dvds 11937  df-gcd 12086  df-prm 12252
This theorem is referenced by:  gausslemma2d  15277
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