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Theorem dchrisum0flblem1 21155
Description: Lemma for dchrisum0flb 21157. Base case, prime power. (Contributed by Mario Carneiro, 5-May-2016.)
Hypotheses
Ref Expression
rpvmasum.z  |-  Z  =  (ℤ/n `  N )
rpvmasum.l  |-  L  =  ( ZRHom `  Z
)
rpvmasum.a  |-  ( ph  ->  N  e.  NN )
rpvmasum2.g  |-  G  =  (DChr `  N )
rpvmasum2.d  |-  D  =  ( Base `  G
)
rpvmasum2.1  |-  .1.  =  ( 0g `  G )
dchrisum0f.f  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
dchrisum0f.x  |-  ( ph  ->  X  e.  D )
dchrisum0flb.r  |-  ( ph  ->  X : ( Base `  Z ) --> RR )
dchrisum0flblem1.1  |-  ( ph  ->  P  e.  Prime )
dchrisum0flblem1.2  |-  ( ph  ->  A  e.  NN0 )
Assertion
Ref Expression
dchrisum0flblem1  |-  ( ph  ->  if ( ( sqr `  ( P ^ A
) )  e.  NN ,  1 ,  0 )  <_  ( F `  ( P ^ A
) ) )
Distinct variable groups:    q, b,
v, A    N, q    P, b, q, v    L, b, v    X, b, v
Allowed substitution hints:    ph( v, q, b)    D( v, q, b)    .1. ( v, q, b)    F( v, q, b)    G( v, q, b)    L( q)    N( v, b)    X( q)    Z( v, q, b)

Proof of Theorem dchrisum0flblem1
Dummy variables  k 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9046 . . . . . 6  |-  1  e.  RR
21a1i 11 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  1  e.  RR )
3 0re 9047 . . . . . 6  |-  0  e.  RR
43a1i 11 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
0  e.  RR )
52, 4ifclda 3726 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR )
61a1i 11 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  e.  RR )
7 fzfid 11267 . . . . . 6  |-  ( ph  ->  ( 0 ... A
)  e.  Fin )
8 dchrisum0flb.r . . . . . . . 8  |-  ( ph  ->  X : ( Base `  Z ) --> RR )
9 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
109nnnn0d 10230 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
11 rpvmasum.z . . . . . . . . . . 11  |-  Z  =  (ℤ/n `  N )
12 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  Z )  =  (
Base `  Z )
13 rpvmasum.l . . . . . . . . . . 11  |-  L  =  ( ZRHom `  Z
)
1411, 12, 13znzrhfo 16783 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
15 fof 5612 . . . . . . . . . 10  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
1610, 14, 153syl 19 . . . . . . . . 9  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
17 dchrisum0flblem1.1 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Prime )
18 prmz 13038 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1917, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  P  e.  ZZ )
2016, 19ffvelrnd 5830 . . . . . . . 8  |-  ( ph  ->  ( L `  P
)  e.  ( Base `  Z ) )
218, 20ffvelrnd 5830 . . . . . . 7  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  RR )
22 elfznn0 11039 . . . . . . 7  |-  ( i  e.  ( 0 ... A )  ->  i  e.  NN0 )
23 reexpcl 11353 . . . . . . 7  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  i  e.  NN0 )  -> 
( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2421, 22, 23syl2an 464 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  e.  RR )
257, 24fsumrecl 12483 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2625adantr 452 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  e.  RR )
27 breq1 4175 . . . . . 6  |-  ( 1  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 1  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
28 breq1 4175 . . . . . 6  |-  ( 0  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 0  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
29 1le1 9606 . . . . . 6  |-  1  <_  1
30 0le1 9507 . . . . . 6  |-  0  <_  1
3127, 28, 29, 30keephyp 3753 . . . . 5  |-  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1
3231a1i 11 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 )
33 dchrisum0flblem1.2 . . . . . . . . . 10  |-  ( ph  ->  A  e.  NN0 )
34 nn0uz 10476 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
3533, 34syl6eleq 2494 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( ZZ>= ` 
0 ) )
36 fzn0 11026 . . . . . . . . 9  |-  ( ( 0 ... A )  =/=  (/)  <->  A  e.  ( ZZ>=
`  0 ) )
3735, 36sylibr 204 . . . . . . . 8  |-  ( ph  ->  ( 0 ... A
)  =/=  (/) )
38 hashnncl 11600 . . . . . . . . 9  |-  ( ( 0 ... A )  e.  Fin  ->  (
( # `  ( 0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
397, 38syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
4037, 39mpbird 224 . . . . . . 7  |-  ( ph  ->  ( # `  (
0 ... A ) )  e.  NN )
4140adantr 452 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  NN )
4241nnge1d 9998 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  (
# `  ( 0 ... A ) ) )
43 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( X `  ( L `  P
) )  =  1 )
4443oveq1d 6055 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( X `  ( L `  P ) ) ^
i )  =  ( 1 ^ i ) )
45 elfzelz 11015 . . . . . . . . 9  |-  ( i  e.  ( 0 ... A )  ->  i  e.  ZZ )
46 1exp 11364 . . . . . . . . 9  |-  ( i  e.  ZZ  ->  (
1 ^ i )  =  1 )
4745, 46syl 16 . . . . . . . 8  |-  ( i  e.  ( 0 ... A )  ->  (
1 ^ i )  =  1 )
4844, 47sylan9eq 2456 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  =  1 )
4948sumeq2dv 12452 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  sum_ i  e.  ( 0 ... A ) 1 )
50 fzfid 11267 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( 0 ... A )  e. 
Fin )
51 ax-1cn 9004 . . . . . . 7  |-  1  e.  CC
52 fsumconst 12528 . . . . . . 7  |-  ( ( ( 0 ... A
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 0 ... A ) 1  =  ( (
# `  ( 0 ... A ) )  x.  1 ) )
5350, 51, 52sylancl 644 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) 1  =  ( ( # `  (
0 ... A ) )  x.  1 ) )
5441nncnd 9972 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  CC )
5554mulid1d 9061 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( # `
 ( 0 ... A ) )  x.  1 )  =  (
# `  ( 0 ... A ) ) )
5649, 53, 553eqtrd 2440 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  (
# `  ( 0 ... A ) ) )
5742, 56breqtrrd 4198 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
585, 6, 26, 32, 57letrd 9183 . . 3  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
59 oveq1 6047 . . . . . . 7  |-  ( 1  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 1  x.  (
1  -  ( X `
 ( L `  P ) ) ) )  =  ( if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  x.  ( 1  -  ( X `  ( L `  P ) ) ) ) )
6059breq1d 4182 . . . . . 6  |-  ( 1  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( ( 1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  <->  ( if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) ) )
61 oveq1 6047 . . . . . . 7  |-  ( 0  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 0  x.  (
1  -  ( X `
 ( L `  P ) ) ) )  =  ( if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  x.  ( 1  -  ( X `  ( L `  P ) ) ) ) )
6261breq1d 4182 . . . . . 6  |-  ( 0  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( ( 0  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  <->  ( if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) ) )
6321adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  RR )
64 resubcl 9321 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( X `  ( L `
 P ) )  e.  RR )  -> 
( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
651, 63, 64sylancr 645 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
6665adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  RR )
6766leidd 9549 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  <_  ( 1  -  ( X `  ( L `  P )
) ) )
6865recnd 9070 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  CC )
6968adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  CC )
7069mulid2d 9062 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  =  ( 1  -  ( X `  ( L `  P ) ) ) )
71 nn0p1nn 10215 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
7233, 71syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN )
7372ad3antrrr 711 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( A  +  1 )  e.  NN )
74730expd 11494 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( 0 ^ ( A  +  1 ) )  =  0 )
75 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( X `  ( L `  P )
)  =  0 )
7675oveq1d 6055 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( 0 ^ ( A  +  1 ) ) )
7774, 76, 753eqtr4d 2446 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
78 neg1cn 10023 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
7933ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  NN0 )
80 expp1 11343 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  A  e.  NN0 )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u 1 ^ A )  x.  -u 1
) )
8178, 79, 80sylancr 645 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u
1 ^ A )  x.  -u 1 ) )
82 prmnn 13037 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P  e.  Prime  ->  P  e.  NN )
8317, 82syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  P  e.  NN )
8483, 33nnexpcld 11499 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( P ^ A
)  e.  NN )
8584nncnd 9972 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( P ^ A
)  e.  CC )
8685ad2antrr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P ^ A )  e.  CC )
8786sqsqrd 12196 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( sqr `  ( P ^ A ) ) ^ 2 )  =  ( P ^ A
) )
8887oveq2d 6056 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( P  pCnt  ( P ^ A ) ) )
8917ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  P  e.  Prime )
90 nnq 10543 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
9190adantl 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
92 nnne0 9988 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
9392adantl 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
94 2z 10268 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
9594a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  ZZ )
96 pcexp 13188 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  Prime  /\  (
( sqr `  ( P ^ A ) )  e.  QQ  /\  ( sqr `  ( P ^ A ) )  =/=  0 )  /\  2  e.  ZZ )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( 2  x.  ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) ) )
9789, 91, 93, 95, 96syl121anc 1189 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( 2  x.  ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) ) )
9879nn0zd 10329 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  ZZ )
99 pcid 13201 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10089, 98, 99syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10188, 97, 1003eqtr3rd 2445 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  =  ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) )
102101oveq2d 6056 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  ( -u 1 ^ ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) ) )
10378a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  -u 1  e.  CC )
104 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  NN )
10589, 104pccld 13179 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  NN0 )
106 2nn0 10194 . . . . . . . . . . . . . . . . 17  |-  2  e.  NN0
107106a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  NN0 )
108103, 105, 107expmuld 11481 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) )  =  ( ( -u 1 ^ 2 ) ^
( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) )
109 sqneg 11397 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  e.  CC  ->  ( -u 1 ^ 2 )  =  ( 1 ^ 2 ) )
11051, 109ax-mp 8 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1 ^ 2 )  =  ( 1 ^ 2 )
111 sq1 11431 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ^ 2 )  =  1
112110, 111eqtri 2424 . . . . . . . . . . . . . . . . 17  |-  ( -u
1 ^ 2 )  =  1
113112oveq1i 6050 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  ( 1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )
114105nn0zd 10329 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  ZZ )
115 1exp 11364 . . . . . . . . . . . . . . . . 17  |-  ( ( P  pCnt  ( sqr `  ( P ^ A
) ) )  e.  ZZ  ->  ( 1 ^ ( P  pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
116114, 115syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
117113, 116syl5eq 2448 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
118102, 108, 1173eqtrd 2440 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  1 )
119118oveq1d 6055 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
12078mulid2i 9049 . . . . . . . . . . . . 13  |-  ( 1  x.  -u 1 )  = 
-u 1
121119, 120syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  -u 1
)
12281, 121eqtrd 2436 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  -u 1 )
123122adantr 452 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( -u 1 ^ ( A  +  1 ) )  =  -u 1
)
12421recnd 9070 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  CC )
125124adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  CC )
126125ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  CC )
127126negnegd 9358 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  ( X `
 ( L `  P ) ) )
128 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  =/=  1
)
129128ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =/=  1 )
130 rpvmasum2.g . . . . . . . . . . . . . . . . . . 19  |-  G  =  (DChr `  N )
131 rpvmasum2.d . . . . . . . . . . . . . . . . . . 19  |-  D  =  ( Base `  G
)
132 dchrisum0f.x . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  X  e.  D )
133132ad3antrrr 711 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  X  e.  D )
134 eqid 2404 . . . . . . . . . . . . . . . . . . 19  |-  (Unit `  Z )  =  (Unit `  Z )
135130, 11, 131, 12, 134, 132, 20dchrn0 20987 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( X `  ( L `  P ) )  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
136135ad2antrr 707 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
)  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
137136biimpa 471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( L `  P
)  e.  (Unit `  Z ) )
138130, 131, 133, 11, 134, 137dchrabs 20997 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  1 )
139 eqeq1 2410 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  ( X `
 ( L `  P ) ) )  =  ( X `  ( L `  P ) )  ->  ( ( abs `  ( X `  ( L `  P ) ) )  =  1  <-> 
( X `  ( L `  P )
)  =  1 ) )
140138, 139syl5ibcom 212 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( abs `  ( X `  ( L `  P ) ) )  =  ( X `  ( L `  P ) )  ->  ( X `  ( L `  P
) )  =  1 ) )
141140necon3ad 2603 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( X `  ( L `  P ) )  =/=  1  ->  -.  ( abs `  ( X `  ( L `  P ) ) )  =  ( X `  ( L `  P ) ) ) )
142129, 141mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -.  ( abs `  ( X `  ( L `  P ) ) )  =  ( X `  ( L `  P ) ) )
14363ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  RR )
144143absord 12173 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( abs `  ( X `  ( L `  P ) ) )  =  ( X `  ( L `  P ) )  \/  ( abs `  ( X `  ( L `  P )
) )  =  -u ( X `  ( L `
 P ) ) ) )
145144ord 367 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( -.  ( abs `  ( X `  ( L `  P )
) )  =  ( X `  ( L `
 P ) )  ->  ( abs `  ( X `  ( L `  P ) ) )  =  -u ( X `  ( L `  P ) ) ) )
146142, 145mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  -u ( X `  ( L `  P ) ) )
147146, 138eqtr3d 2438 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u ( X `  ( L `  P )
)  =  1 )
148147negeqd 9256 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  -u 1
)
149127, 148eqtr3d 2438 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =  -u 1
)
150149oveq1d 6055 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( -u 1 ^ ( A  + 
1 ) ) )
151123, 150, 1493eqtr4d 2446 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
15277, 151pm2.61dane 2645 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
153152oveq2d 6056 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  =  ( 1  -  ( X `  ( L `  P )
) ) )
15467, 70, 1533brtr4d 4202 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
15568mul02d 9220 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  =  0 )
156 peano2nn0 10216 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
15733, 156syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
15821, 157reexpcld 11495 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )
159158adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  RR )
160159recnd 9070 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  CC )
161160abscld 12193 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  e.  RR )
1621a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  e.  RR )
163159leabsd 12172 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  ( abs `  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
164157adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e. 
NN0 )
165125, 164absexpd 12209 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  =  ( ( abs `  ( X `
 ( L `  P ) ) ) ^ ( A  + 
1 ) ) )
166125abscld 12193 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  e.  RR )
167125absge0d 12201 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( abs `  ( X `
 ( L `  P ) ) ) )
168130, 131, 11, 12, 132, 20dchrabs2 20999 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( X `  ( L `  P ) ) )  <_  1 )
169168adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  <_  1
)
170 exple1 11394 . . . . . . . . . . . 12  |-  ( ( ( ( abs `  ( X `  ( L `  P ) ) )  e.  RR  /\  0  <_  ( abs `  ( X `  ( L `  P ) ) )  /\  ( abs `  ( X `  ( L `  P ) ) )  <_  1 )  /\  ( A  +  1
)  e.  NN0 )  ->  ( ( abs `  ( X `  ( L `  P ) ) ) ^ ( A  + 
1 ) )  <_ 
1 )
171166, 167, 169, 164, 170syl31anc 1187 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( abs `  ( X `  ( L `  P ) ) ) ^ ( A  +  1 ) )  <_  1 )
172165, 171eqbrtrd 4192 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <_  1 )
173159, 161, 162, 163, 172letrd 9183 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  1
)
174 subge0 9497 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 0  <_  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  <-> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  <_  1 ) )
1751, 159, 174sylancr 645 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  <_  ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <->  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) )  <_  1
) )
176173, 175mpbird 224 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( 1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) ) )
177155, 176eqbrtrd 4192 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
178177adantr 452 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
( 0  x.  (
1  -  ( X `
 ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
17960, 62, 154, 178ifbothda 3729 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
1801, 3keepel 3756 . . . . . . 7  |-  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR
181180a1i 11 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR )
182 resubcl 9321 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) )  e.  RR )
1831, 159, 182sylancr 645 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  e.  RR )
184128necomd 2650 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  =/=  ( X `  ( L `
 P ) ) )
18563leabsd 12172 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  ( abs `  ( X `  ( L `  P ) ) ) )
18663, 166, 162, 185, 169letrd 9183 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  1
)
18763, 162, 186leltned 9180 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  1  =/=  ( X `  ( L `  P ) ) ) )
188184, 187mpbird 224 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <  1
)
189 posdif 9477 . . . . . . . 8  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  1  e.  RR )  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  ( 1  -  ( X `  ( L `  P )
) ) ) )
19063, 1, 189sylancl 644 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  (
1  -  ( X `
 ( L `  P ) ) ) ) )
191188, 190mpbid 202 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <  ( 1  -  ( X `
 ( L `  P ) ) ) )
192 lemuldiv 9845 . . . . . 6  |-  ( ( if ( ( sqr `  ( P ^ A
) )  e.  NN ,  1 ,  0 )  e.  RR  /\  ( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) )  e.  RR  /\  ( ( 1  -  ( X `  ( L `  P )
) )  e.  RR  /\  0  <  ( 1  -  ( X `  ( L `  P ) ) ) ) )  ->  ( ( if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  <->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
( ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  /  ( 1  -  ( X `  ( L `  P ) ) ) ) ) )
193181, 183, 65, 191, 192syl112anc 1188 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  <->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
( ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  /  ( 1  -  ( X `  ( L `  P ) ) ) ) ) )
194179, 193mpbid 202 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
( ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  /  ( 1  -  ( X `  ( L `  P ) ) ) ) )
19533nn0zd 10329 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
196 fzval3 11135 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
197195, 196syl 16 . . . . . . 7  |-  ( ph  ->  ( 0 ... A
)  =  ( 0..^ ( A  +  1 ) ) )
198197adantr 452 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
199198sumeq1d 12450 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  sum_ i  e.  ( 0..^ ( A  +  1 ) ) ( ( X `  ( L `
 P ) ) ^ i ) )
200 0nn0 10192 . . . . . . 7  |-  0  e.  NN0
201200a1i 11 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  e.  NN0 )
202157, 34syl6eleq 2494 . . . . . . 7  |-  ( ph  ->  ( A  +  1 )  e.  ( ZZ>= ` 
0 ) )
203202adantr 452 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e.  ( ZZ>= `  0 )
)
204125, 128, 201, 203geoserg 12600 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  sum_ i  e.  ( 0..^ ( A  +  1 ) ) ( ( X `  ( L `  P ) ) ^ i )  =  ( ( ( ( X `  ( L `  P )
) ^ 0 )  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  / 
( 1  -  ( X `  ( L `  P ) ) ) ) )
205125exp0d 11472 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
0 )  =  1 )
206205oveq1d 6055 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( (
( X `  ( L `  P )
) ^ 0 )  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  =  ( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
207206oveq1d 6055 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( (
( ( X `  ( L `  P ) ) ^ 0 )  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  / 
( 1  -  ( X `  ( L `  P ) ) ) )  =  ( ( 1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  /  ( 1  -  ( X `  ( L `  P )
) ) ) )
208199, 204, 2073eqtrd 2440 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  ( ( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) )  /  ( 1  -  ( X `  ( L `  P ) ) ) ) )
209194, 208breqtrrd 4198 . . 3  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
21058, 209pm2.61dane 2645 . 2  |-  ( ph  ->  if ( ( sqr `  ( P ^ A
) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
211 rpvmasum2.1 . . . . 5  |-  .1.  =  ( 0g `  G )
212 dchrisum0f.f . . . . 5  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
21311, 13, 9, 130, 131, 211, 212dchrisum0fval 21152 . . . 4  |-  ( ( P ^ A )  e.  NN  ->  ( F `  ( P ^ A ) )  = 
sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
21484, 213syl 16 . . 3  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
215 fveq2 5687 . . . . 5  |-  ( k  =  ( P ^
i )  ->  ( L `  k )  =  ( L `  ( P ^ i ) ) )
216215fveq2d 5691 . . . 4  |-  ( k  =  ( P ^
i )  ->  ( X `  ( L `  k ) )  =  ( X `  ( L `  ( P ^ i ) ) ) )
217 eqid 2404 . . . . . 6  |-  ( b  e.  ( 0 ... A )  |->  ( P ^ b ) )  =  ( b  e.  ( 0 ... A
)  |->  ( P ^
b ) )
218217dvdsppwf1o 20924 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
b  e.  ( 0 ... A )  |->  ( P ^ b ) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  |  q 
||  ( P ^ A ) } )
21917, 33, 218syl2anc 643 . . . 4  |-  ( ph  ->  ( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  | 
q  ||  ( P ^ A ) } )
220 oveq2 6048 . . . . . 6  |-  ( b  =  i  ->  ( P ^ b )  =  ( P ^ i
) )
221 ovex 6065 . . . . . 6  |-  ( P ^ b )  e. 
_V
222220, 217, 221fvmpt3i 5768 . . . . 5  |-  ( i  e.  ( 0 ... A )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
223222adantl 453 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
2248adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  X : (
Base `  Z ) --> RR )
225 elrabi 3050 . . . . . . . 8  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  NN )
226225nnzd 10330 . . . . . . 7  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  ZZ )
227 ffvelrn 5827 . . . . . . 7  |-  ( ( L : ZZ --> ( Base `  Z )  /\  k  e.  ZZ )  ->  ( L `  k )  e.  ( Base `  Z
) )
22816, 226, 227syl2an 464 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( L `  k )  e.  (
Base `  Z )
)
229224, 228ffvelrnd 5830 . . . . 5  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  RR )
230229recnd 9070 . . . 4  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  CC )
231216, 7, 219, 223, 230fsumf1o 12472 . . 3  |-  ( ph  -> 
sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) )  = 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) ) )
232 zsubrg 16707 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
233 eqid 2404 . . . . . . . . . . . 12  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
234233subrgsubm 15836 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
235232, 234mp1i 12 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
23622adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  i  e.  NN0 )
23719adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  P  e.  ZZ )
238 eqid 2404 . . . . . . . . . . 11  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
239 cnfldex 16661 . . . . . . . . . . . . 13  |-fld  e.  _V
240 zex 10247 . . . . . . . . . . . . 13  |-  ZZ  e.  _V
241 eqid 2404 . . . . . . . . . . . . . 14  |-  (flds  ZZ )  =  (flds  ZZ )
242241, 233mgpress 15614 . . . . . . . . . . . . 13  |-  ( (fld  e. 
_V  /\  ZZ  e.  _V )  ->  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp `  (flds  ZZ ) ) )
243239, 240, 242mp2an 654 . . . . . . . . . . . 12  |-  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp `  (flds  ZZ ) )
244243eqcomi 2408 . . . . . . . . . . 11  |-  (mulGrp `  (flds  ZZ ) )  =  ( (mulGrp ` fld )s  ZZ )
245 eqid 2404 . . . . . . . . . . 11  |-  (.g `  (mulGrp `  (flds  ZZ ) ) )  =  (.g `  (mulGrp `  (flds  ZZ )
) )
246238, 244, 245submmulg 14880 . . . . . . . . . 10  |-  ( ( ZZ  e.  (SubMnd `  (mulGrp ` fld ) )  /\  i  e.  NN0  /\  P  e.  ZZ )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp `  (flds  ZZ ) ) ) P ) )
247235, 236, 237, 246syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp `  (flds  ZZ ) ) ) P ) )
24883nncnd 9972 . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
249 cnfldexp 16689 . . . . . . . . . 10  |-  ( ( P  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^ i
) )
250248, 22, 249syl2an 464 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^
i ) )
251247, 250eqtr3d 2438 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp `  (flds  ZZ )
) ) P )  =  ( P ^
i ) )
252251fveq2d 5691 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp `  (flds  ZZ )
) ) P ) )  =  ( L `
 ( P ^
i ) ) )
25311zncrng 16780 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
254 crngrng 15629 . . . . . . . . . . 11  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
25510, 253, 2543syl 19 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  Ring )
256241, 13zrhrhm 16748 . . . . . . . . . 10  |-  ( Z  e.  Ring  ->  L  e.  ( (flds  ZZ ) RingHom  Z ) )
257 eqid 2404 . . . . . . . . . . 11  |-  (mulGrp `  (flds  ZZ ) )  =  (mulGrp `  (flds  ZZ ) )
258 eqid 2404 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
259257, 258rhmmhm 15780 . . . . . . . . . 10  |-  ( L  e.  ( (flds  ZZ ) RingHom  Z )  ->  L  e.  ( (mulGrp `  (flds  ZZ ) ) MndHom  (mulGrp `  Z ) ) )
260255, 256, 2593syl 19 . . . . . . . . 9  |-  ( ph  ->  L  e.  ( (mulGrp `  (flds  ZZ ) ) MndHom  (mulGrp `  Z ) ) )
261260adantr 452 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  L  e.  ( (mulGrp `  (flds  ZZ )
) MndHom  (mulGrp `  Z )
) )
262 subrgsubg 15829 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
263241subgbas 14903 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubGrp ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
264232, 262, 263mp2b 10 . . . . . . . . . 10  |-  ZZ  =  ( Base `  (flds  ZZ ) )
265257, 264mgpbas 15609 . . . . . . . . 9  |-  ZZ  =  ( Base `  (mulGrp `  (flds  ZZ )
) )
266 eqid 2404 . . . . . . . . 9  |-  (.g `  (mulGrp `  Z ) )  =  (.g `  (mulGrp `  Z
) )
267265, 245, 266mhmmulg 14877 . . . . . . . 8  |-  ( ( L  e.  ( (mulGrp `  (flds  ZZ ) ) MndHom  (mulGrp `  Z ) )  /\  i  e.  NN0  /\  P  e.  ZZ )  ->  ( L `  ( i
(.g `  (mulGrp `  (flds  ZZ )
) ) P ) )  =  ( i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )
268261, 236, 237, 267syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp `  (flds  ZZ )
) ) P ) )  =  ( i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )
269252, 268eqtr3d 2438 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( P ^ i ) )  =  ( i (.g `  (mulGrp `  Z )
) ( L `  P ) ) )
270269fveq2d 5691 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( X `  (
i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) ) )
271130, 11, 131dchrmhm 20978 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
272271, 132sseldi 3306 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
273272adantr 452 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
27420adantr 452 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  P )  e.  ( Base `  Z
) )
275258, 12mgpbas 15609 . . . . . . 7  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
276275, 266, 238mhmmulg 14877 . . . . . 6  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  i  e.  NN0 
/\  ( L `  P )  e.  (
Base `  Z )
)  ->  ( X `  ( i (.g `  (mulGrp `  Z ) ) ( L `  P ) ) )  =  ( i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) ) )
277273, 236, 274, 276syl3anc 1184 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( i
(.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )  =  ( i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) ) )
278 cnfldexp 16689 . . . . . 6  |-  ( ( ( X `  ( L `  P )
)  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) ( X `  ( L `  P ) ) )  =  ( ( X `  ( L `  P )
) ^ i ) )
279124, 22, 278syl2an 464 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) )  =  ( ( X `
 ( L `  P ) ) ^
i ) )
280270, 277, 2793eqtrd 2440 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( ( X `  ( L `  P ) ) ^ i ) )
281280sumeq2dv 12452 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) )  =  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
282214, 231, 2813eqtrd 2440 . 2  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i ) )
283210, 282breqtrrd 4198 1  |-  ( ph  ->  if ( ( sqr `  ( P ^ A
) )  e.  NN ,  1 ,  0 )  <_  ( F `  ( P ^ A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916   (/)c0 3588   ifcif 3699   class class class wbr 4172    e. cmpt 4226   -->wf 5409   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   QQcq 10530   ...cfz 10999  ..^cfzo 11090   ^cexp 11337   #chash 11573   sqrcsqr 11993   abscabs 11994   sum_csu 12434    || cdivides 12807   Primecprime 13034    pCnt cpc 13165   Basecbs 13424   ↾s cress 13425   0gc0g 13678  .gcmg 14644   MndHom cmhm 14691  SubMndcsubmnd 14692  SubGrpcsubg 14893  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616  Unitcui 15699   RingHom crh 15772  SubRingcsubrg 15819  ℂfldccnfld 16658   ZRHomczrh 16733  ℤ/nczn 16736  DChrcdchr 20969
This theorem is referenced by:  dchrisum0flblem2  21156  dchrisum0flb  21157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-divs 13690  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-nsg 14897  df-eqg 14898  df-ghm 14959  df-cntz 15071  df-od 15122  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-zrh 16737  df-zn 16740  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-dchr 20970
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