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Theorem dchrisum0flblem1 21204
Description: Lemma for dchrisum0flb 21206. 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 9092 . . . . . 6  |-  1  e.  RR
21a1i 11 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  1  e.  RR )
3 0re 9093 . . . . . 6  |-  0  e.  RR
43a1i 11 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
0  e.  RR )
52, 4ifclda 3768 . . . 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 11314 . . . . . 6  |-  ( ph  ->  ( 0 ... A
)  e.  Fin )
8 dchrisum0flb.r . . . . . . . 8  |-  ( ph  ->  X : ( Base `  Z ) --> RR )
9 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
109nnnn0d 10276 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
11 rpvmasum.z . . . . . . . . . . 11  |-  Z  =  (ℤ/n `  N )
12 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  Z )  =  (
Base `  Z )
13 rpvmasum.l . . . . . . . . . . 11  |-  L  =  ( ZRHom `  Z
)
1411, 12, 13znzrhfo 16830 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
15 fof 5655 . . . . . . . . . 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 13085 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1917, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  P  e.  ZZ )
2016, 19ffvelrnd 5873 . . . . . . . 8  |-  ( ph  ->  ( L `  P
)  e.  ( Base `  Z ) )
218, 20ffvelrnd 5873 . . . . . . 7  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  RR )
22 elfznn0 11085 . . . . . . 7  |-  ( i  e.  ( 0 ... A )  ->  i  e.  NN0 )
23 reexpcl 11400 . . . . . . 7  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  i  e.  NN0 )  -> 
( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2421, 22, 23syl2an 465 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  e.  RR )
257, 24fsumrecl 12530 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2625adantr 453 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  e.  RR )
27 breq1 4217 . . . . . 6  |-  ( 1  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 1  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
28 breq1 4217 . . . . . 6  |-  ( 0  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 0  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
29 1le1 9652 . . . . . 6  |-  1  <_  1
30 0le1 9553 . . . . . 6  |-  0  <_  1
3127, 28, 29, 30keephyp 3795 . . . . 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 10522 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
3533, 34syl6eleq 2528 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( ZZ>= ` 
0 ) )
36 fzn0 11072 . . . . . . . . 9  |-  ( ( 0 ... A )  =/=  (/)  <->  A  e.  ( ZZ>=
`  0 ) )
3735, 36sylibr 205 . . . . . . . 8  |-  ( ph  ->  ( 0 ... A
)  =/=  (/) )
38 hashnncl 11647 . . . . . . . . 9  |-  ( ( 0 ... A )  e.  Fin  ->  (
( # `  ( 0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
397, 38syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
4037, 39mpbird 225 . . . . . . 7  |-  ( ph  ->  ( # `  (
0 ... A ) )  e.  NN )
4140adantr 453 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  NN )
4241nnge1d 10044 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  (
# `  ( 0 ... A ) ) )
43 simpr 449 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( X `  ( L `  P
) )  =  1 )
4443oveq1d 6098 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( X `  ( L `  P ) ) ^
i )  =  ( 1 ^ i ) )
45 elfzelz 11061 . . . . . . . . 9  |-  ( i  e.  ( 0 ... A )  ->  i  e.  ZZ )
46 1exp 11411 . . . . . . . . 9  |-  ( i  e.  ZZ  ->  (
1 ^ i )  =  1 )
4745, 46syl 16 . . . . . . . 8  |-  ( i  e.  ( 0 ... A )  ->  (
1 ^ i )  =  1 )
4844, 47sylan9eq 2490 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  =  1 )
4948sumeq2dv 12499 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  sum_ i  e.  ( 0 ... A ) 1 )
50 fzfid 11314 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( 0 ... A )  e. 
Fin )
51 ax-1cn 9050 . . . . . . 7  |-  1  e.  CC
52 fsumconst 12575 . . . . . . 7  |-  ( ( ( 0 ... A
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 0 ... A ) 1  =  ( (
# `  ( 0 ... A ) )  x.  1 ) )
5350, 51, 52sylancl 645 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) 1  =  ( ( # `  (
0 ... A ) )  x.  1 ) )
5441nncnd 10018 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  CC )
5554mulid1d 9107 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( # `
 ( 0 ... A ) )  x.  1 )  =  (
# `  ( 0 ... A ) ) )
5649, 53, 553eqtrd 2474 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  (
# `  ( 0 ... A ) ) )
5742, 56breqtrrd 4240 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
585, 6, 26, 32, 57letrd 9229 . . 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 6090 . . . . . . 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 4224 . . . . . 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 6090 . . . . . . 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 4224 . . . . . 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 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  RR )
64 resubcl 9367 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( X `  ( L `
 P ) )  e.  RR )  -> 
( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
651, 63, 64sylancr 646 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
6665adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  RR )
6766leidd 9595 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  <_  ( 1  -  ( X `  ( L `  P )
) ) )
6865recnd 9116 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  CC )
6968adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  CC )
7069mulid2d 9108 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  =  ( 1  -  ( X `  ( L `  P ) ) ) )
71 nn0p1nn 10261 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
7233, 71syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN )
7372ad3antrrr 712 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( A  +  1 )  e.  NN )
74730expd 11541 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( 0 ^ ( A  +  1 ) )  =  0 )
75 simpr 449 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( X `  ( L `  P )
)  =  0 )
7675oveq1d 6098 . . . . . . . . . 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 2480 . . . . . . . . 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 10069 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
7933ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  NN0 )
80 expp1 11390 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  A  e.  NN0 )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u 1 ^ A )  x.  -u 1
) )
8178, 79, 80sylancr 646 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u
1 ^ A )  x.  -u 1 ) )
82 prmnn 13084 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P  e.  Prime  ->  P  e.  NN )
8317, 82syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  P  e.  NN )
8483, 33nnexpcld 11546 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( P ^ A
)  e.  NN )
8584nncnd 10018 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( P ^ A
)  e.  CC )
8685ad2antrr 708 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P ^ A )  e.  CC )
8786sqsqrd 12243 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( sqr `  ( P ^ A ) ) ^ 2 )  =  ( P ^ A
) )
8887oveq2d 6099 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( P  pCnt  ( P ^ A ) ) )
8917ad2antrr 708 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  P  e.  Prime )
90 nnq 10589 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
9190adantl 454 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
92 nnne0 10034 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
9392adantl 454 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
94 2z 10314 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
9594a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  ZZ )
96 pcexp 13235 . . . . . . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . . . . . 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 10375 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  ZZ )
99 pcid 13248 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10089, 98, 99syl2anc 644 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10188, 97, 1003eqtr3rd 2479 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  =  ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) )
102101oveq2d 6099 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  NN )
10589, 104pccld 13226 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  NN0 )
106 2nn0 10240 . . . . . . . . . . . . . . . . 17  |-  2  e.  NN0
107106a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  NN0 )
108103, 105, 107expmuld 11528 . . . . . . . . . . . . . . 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 11444 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  e.  CC  ->  ( -u 1 ^ 2 )  =  ( 1 ^ 2 ) )
11051, 109ax-mp 8 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1 ^ 2 )  =  ( 1 ^ 2 )
111 sq1 11478 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ^ 2 )  =  1
112110, 111eqtri 2458 . . . . . . . . . . . . . . . . 17  |-  ( -u
1 ^ 2 )  =  1
113112oveq1i 6093 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  ( 1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )
114105nn0zd 10375 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  ZZ )
115 1exp 11411 . . . . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
118102, 108, 1173eqtrd 2474 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  1 )
119118oveq1d 6098 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
12078mulid2i 9095 . . . . . . . . . . . . 13  |-  ( 1  x.  -u 1 )  = 
-u 1
121119, 120syl6eq 2486 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  -u 1
)
12281, 121eqtrd 2470 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  -u 1 )
123122adantr 453 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( -u 1 ^ ( A  +  1 ) )  =  -u 1
)
12421recnd 9116 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  CC )
125124adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  CC )
126125ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  CC )
127126negnegd 9404 . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  =/=  1
)
129128ad2antrr 708 . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  X  e.  D )
134 eqid 2438 . . . . . . . . . . . . . . . . . . 19  |-  (Unit `  Z )  =  (Unit `  Z )
135130, 11, 131, 12, 134, 132, 20dchrn0 21036 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( X `  ( L `  P ) )  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
136135ad2antrr 708 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
)  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
137136biimpa 472 . . . . . . . . . . . . . . . . . . 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 21046 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  1 )
139 eqeq1 2444 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  ( X `
 ( L `  P ) ) )  =  ( X `  ( L `  P ) )  ->  ( ( abs `  ( X `  ( L `  P ) ) )  =  1  <-> 
( X `  ( L `  P )
)  =  1 ) )
140138, 139syl5ibcom 213 . . . . . . . . . . . . . . . . 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 2639 . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  RR )
144143absord 12220 . . . . . . . . . . . . . . . 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 368 . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u ( X `  ( L `  P )
)  =  1 )
148147negeqd 9302 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  -u 1
)
149127, 148eqtr3d 2472 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =  -u 1
)
150149oveq1d 6098 . . . . . . . . . 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 2480 . . . . . . . . 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 2684 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
153152oveq2d 6099 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  =  ( 1  -  ( X `  ( L `  P )
) ) )
15467, 70, 1533brtr4d 4244 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
15568mul02d 9266 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  =  0 )
156 peano2nn0 10262 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
15733, 156syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
15821, 157reexpcld 11542 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )
159158adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  RR )
160159recnd 9116 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  CC )
161160abscld 12240 . . . . . . . . . 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 12219 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  ( abs `  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
164157adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e. 
NN0 )
165125, 164absexpd 12256 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  =  ( ( abs `  ( X `
 ( L `  P ) ) ) ^ ( A  + 
1 ) ) )
166125abscld 12240 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  e.  RR )
167125absge0d 12248 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( abs `  ( X `
 ( L `  P ) ) ) )
168130, 131, 11, 12, 132, 20dchrabs2 21048 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( X `  ( L `  P ) ) )  <_  1 )
169168adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  <_  1
)
170 exple1 11441 . . . . . . . . . . . 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 1188 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( abs `  ( X `  ( L `  P ) ) ) ^ ( A  +  1 ) )  <_  1 )
172165, 171eqbrtrd 4234 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <_  1 )
173159, 161, 162, 163, 172letrd 9229 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  1
)
174 subge0 9543 . . . . . . . . . 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 646 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  <_  ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <->  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) )  <_  1
) )
176173, 175mpbird 225 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( 1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) ) )
177155, 176eqbrtrd 4234 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
178177adantr 453 . . . . . 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 3771 . . . . 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 3798 . . . . . . 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 9367 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) )  e.  RR )
1831, 159, 182sylancr 646 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  e.  RR )
184128necomd 2689 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  =/=  ( X `  ( L `
 P ) ) )
18563leabsd 12219 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  ( abs `  ( X `  ( L `  P ) ) ) )
18663, 166, 162, 185, 169letrd 9229 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  1
)
18763, 162, 186leltned 9226 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  1  =/=  ( X `  ( L `  P ) ) ) )
188184, 187mpbird 225 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <  1
)
189 posdif 9523 . . . . . . . 8  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  1  e.  RR )  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  ( 1  -  ( X `  ( L `  P )
) ) ) )
19063, 1, 189sylancl 645 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  (
1  -  ( X `
 ( L `  P ) ) ) ) )
191188, 190mpbid 203 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <  ( 1  -  ( X `
 ( L `  P ) ) ) )
192 lemuldiv 9891 . . . . . 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 1189 . . . . 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 203 . . . 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 10375 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
196 fzval3 11182 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
197195, 196syl 16 . . . . . . 7  |-  ( ph  ->  ( 0 ... A
)  =  ( 0..^ ( A  +  1 ) ) )
198197adantr 453 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
199198sumeq1d 12497 . . . . 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 10238 . . . . . . 7  |-  0  e.  NN0
201200a1i 11 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  e.  NN0 )
202157, 34syl6eleq 2528 . . . . . . 7  |-  ( ph  ->  ( A  +  1 )  e.  ( ZZ>= ` 
0 ) )
203202adantr 453 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e.  ( ZZ>= `  0 )
)
204125, 128, 201, 203geoserg 12647 . . . . 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 11519 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
0 )  =  1 )
206205oveq1d 6098 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( (
( X `  ( L `  P )
) ^ 0 )  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  =  ( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
207206oveq1d 6098 . . . . 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 2474 . . . 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 4240 . . 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 2684 . 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 21201 . . . 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 5730 . . . . 5  |-  ( k  =  ( P ^
i )  ->  ( L `  k )  =  ( L `  ( P ^ i ) ) )
216215fveq2d 5734 . . . 4  |-  ( k  =  ( P ^
i )  ->  ( X `  ( L `  k ) )  =  ( X `  ( L `  ( P ^ i ) ) ) )
217 eqid 2438 . . . . . 6  |-  ( b  e.  ( 0 ... A )  |->  ( P ^ b ) )  =  ( b  e.  ( 0 ... A
)  |->  ( P ^
b ) )
218217dvdsppwf1o 20973 . . . . 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 644 . . . 4  |-  ( ph  ->  ( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  | 
q  ||  ( P ^ A ) } )
220 oveq2 6091 . . . . . 6  |-  ( b  =  i  ->  ( P ^ b )  =  ( P ^ i
) )
221 ovex 6108 . . . . . 6  |-  ( P ^ b )  e. 
_V
222220, 217, 221fvmpt3i 5811 . . . . 5  |-  ( i  e.  ( 0 ... A )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
223222adantl 454 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
2248adantr 453 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  X : (
Base `  Z ) --> RR )
225 elrabi 3092 . . . . . . . 8  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  NN )
226225nnzd 10376 . . . . . . 7  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  ZZ )
227 ffvelrn 5870 . . . . . . 7  |-  ( ( L : ZZ --> ( Base `  Z )  /\  k  e.  ZZ )  ->  ( L `  k )  e.  ( Base `  Z
) )
22816, 226, 227syl2an 465 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( L `  k )  e.  (
Base `  Z )
)
229224, 228ffvelrnd 5873 . . . . 5  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  RR )
230229recnd 9116 . . . 4  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  CC )
231216, 7, 219, 223, 230fsumf1o 12519 . . 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 16754 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
233 eqid 2438 . . . . . . . . . . . 12  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
234233subrgsubm 15883 . . . . . . . . . . 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 454 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  i  e.  NN0 )
23719adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  P  e.  ZZ )
238 eqid 2438 . . . . . . . . . . 11  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
239 cnfldex 16708 . . . . . . . . . . . . 13  |-fld  e.  _V
240 zex 10293 . . . . . . . . . . . . 13  |-  ZZ  e.  _V
241 eqid 2438 . . . . . . . . . . . . . 14  |-  (flds  ZZ )  =  (flds  ZZ )
242241, 233mgpress 15661 . . . . . . . . . . . . 13  |-  ( (fld  e. 
_V  /\  ZZ  e.  _V )  ->  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp `  (flds  ZZ ) ) )
243239, 240, 242mp2an 655 . . . . . . . . . . . 12  |-  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp `  (flds  ZZ ) )
244243eqcomi 2442 . . . . . . . . . . 11  |-  (mulGrp `  (flds  ZZ ) )  =  ( (mulGrp ` fld )s  ZZ )
245 eqid 2438 . . . . . . . . . . 11  |-  (.g `  (mulGrp `  (flds  ZZ ) ) )  =  (.g `  (mulGrp `  (flds  ZZ )
) )
246238, 244, 245submmulg 14927 . . . . . . . . . 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 1185 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp `  (flds  ZZ ) ) ) P ) )
24883nncnd 10018 . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
249 cnfldexp 16736 . . . . . . . . . 10  |-  ( ( P  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^ i
) )
250248, 22, 249syl2an 465 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^
i ) )
251247, 250eqtr3d 2472 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp `  (flds  ZZ )
) ) P )  =  ( P ^
i ) )
252251fveq2d 5734 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp `  (flds  ZZ )
) ) P ) )  =  ( L `
 ( P ^
i ) ) )
25311zncrng 16827 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
254 crngrng 15676 . . . . . . . . . . 11  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
25510, 253, 2543syl 19 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  Ring )
256241, 13zrhrhm 16795 . . . . . . . . . 10  |-  ( Z  e.  Ring  ->  L  e.  ( (flds  ZZ ) RingHom  Z ) )
257 eqid 2438 . . . . . . . . . . 11  |-  (mulGrp `  (flds  ZZ ) )  =  (mulGrp `  (flds  ZZ ) )
258 eqid 2438 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
259257, 258rhmmhm 15827 . . . . . . . . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  L  e.  ( (mulGrp `  (flds  ZZ )
) MndHom  (mulGrp `  Z )
) )
262 subrgsubg 15876 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
263241subgbas 14950 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubGrp ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
264232, 262, 263mp2b 10 . . . . . . . . . 10  |-  ZZ  =  ( Base `  (flds  ZZ ) )
265257, 264mgpbas 15656 . . . . . . . . 9  |-  ZZ  =  ( Base `  (mulGrp `  (flds  ZZ )
) )
266 eqid 2438 . . . . . . . . 9  |-  (.g `  (mulGrp `  Z ) )  =  (.g `  (mulGrp `  Z
) )
267265, 245, 266mhmmulg 14924 . . . . . . . 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 1185 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp `  (flds  ZZ )
) ) P ) )  =  ( i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )
269252, 268eqtr3d 2472 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( P ^ i ) )  =  ( i (.g `  (mulGrp `  Z )
) ( L `  P ) ) )
270269fveq2d 5734 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( X `  (
i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) ) )
271130, 11, 131dchrmhm 21027 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
272271, 132sseldi 3348 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
273272adantr 453 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
27420adantr 453 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  P )  e.  ( Base `  Z
) )
275258, 12mgpbas 15656 . . . . . . 7  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
276275, 266, 238mhmmulg 14924 . . . . . 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 1185 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( i
(.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )  =  ( i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) ) )
278 cnfldexp 16736 . . . . . 6  |-  ( ( ( X `  ( L `  P )
)  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) ( X `  ( L `  P ) ) )  =  ( ( X `  ( L `  P )
) ^ i ) )
279124, 22, 278syl2an 465 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) )  =  ( ( X `
 ( L `  P ) ) ^
i ) )
280270, 277, 2793eqtrd 2474 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( ( X `  ( L `  P ) ) ^ i ) )
281280sumeq2dv 12499 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) )  =  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
282214, 231, 2813eqtrd 2474 . 2  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i ) )
283210, 282breqtrrd 4240 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   _Vcvv 2958   (/)c0 3630   ifcif 3741   class class class wbr 4214    e. cmpt 4268   -->wf 5452   -onto->wfo 5454   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   Fincfn 7111   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    < clt 9122    <_ cle 9123    - cmin 9293   -ucneg 9294    / cdiv 9679   NNcn 10002   2c2 10051   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   QQcq 10576   ...cfz 11045  ..^cfzo 11137   ^cexp 11384   #chash 11620   sqrcsqr 12040   abscabs 12041   sum_csu 12481    || cdivides 12854   Primecprime 13081    pCnt cpc 13212   Basecbs 13471   ↾s cress 13472   0gc0g 13725  .gcmg 14691   MndHom cmhm 14738  SubMndcsubmnd 14739  SubGrpcsubg 14940  mulGrpcmgp 15650   Ringcrg 15662   CRingccrg 15663  Unitcui 15746   RingHom crh 15819  SubRingcsubrg 15866  ℂfldccnfld 16705   ZRHomczrh 16780  ℤ/nczn 16783  DChrcdchr 21018
This theorem is referenced by:  dchrisum0flblem2  21205  dchrisum0flb  21206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-pi 12677  df-dvds 12855  df-gcd 13009  df-prm 13082  df-pc 13213  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-divs 13737  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-nsg 14944  df-eqg 14945  df-ghm 15006  df-cntz 15118  df-od 15169  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-rnghom 15821  df-drng 15839  df-subrg 15868  df-lmod 15954  df-lss 16011  df-lsp 16050  df-sra 16246  df-rgmod 16247  df-lidl 16248  df-rsp 16249  df-2idl 16305  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-zrh 16784  df-zn 16787  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456  df-cxp 20457  df-dchr 21019
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