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Theorem dchrisum0flblem1 20657
Description: Lemma for dchrisum0flb 20659. 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 8837 . . . . . 6  |-  1  e.  RR
21a1i 10 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  1  e.  RR )
3 0re 8838 . . . . . 6  |-  0  e.  RR
43a1i 10 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
0  e.  RR )
52, 4ifclda 3592 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR )
61a1i 10 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  e.  RR )
7 fzfid 11035 . . . . . 6  |-  ( ph  ->  ( 0 ... A
)  e.  Fin )
8 dchrisum0flb.r . . . . . . . 8  |-  ( ph  ->  X : ( Base `  Z ) --> RR )
9 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
109nnnn0d 10018 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
11 rpvmasum.z . . . . . . . . . . 11  |-  Z  =  (ℤ/n `  N )
12 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  Z )  =  (
Base `  Z )
13 rpvmasum.l . . . . . . . . . . 11  |-  L  =  ( ZRHom `  Z
)
1411, 12, 13znzrhfo 16501 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
15 fof 5451 . . . . . . . . . 10  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
1610, 14, 153syl 18 . . . . . . . . 9  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
17 dchrisum0flblem1.1 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Prime )
18 prmz 12762 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1917, 18syl 15 . . . . . . . . 9  |-  ( ph  ->  P  e.  ZZ )
20 ffvelrn 5663 . . . . . . . . 9  |-  ( ( L : ZZ --> ( Base `  Z )  /\  P  e.  ZZ )  ->  ( L `  P )  e.  ( Base `  Z
) )
2116, 19, 20syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( L `  P
)  e.  ( Base `  Z ) )
22 ffvelrn 5663 . . . . . . . 8  |-  ( ( X : ( Base `  Z ) --> RR  /\  ( L `  P )  e.  ( Base `  Z
) )  ->  ( X `  ( L `  P ) )  e.  RR )
238, 21, 22syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  RR )
24 elfznn0 10822 . . . . . . 7  |-  ( i  e.  ( 0 ... A )  ->  i  e.  NN0 )
25 reexpcl 11120 . . . . . . 7  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  i  e.  NN0 )  -> 
( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2623, 24, 25syl2an 463 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  e.  RR )
277, 26fsumrecl 12207 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2827adantr 451 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  e.  RR )
29 breq1 4026 . . . . . 6  |-  ( 1  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 1  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
30 breq1 4026 . . . . . 6  |-  ( 0  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 0  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
31 1le1 9396 . . . . . 6  |-  1  <_  1
32 0le1 9297 . . . . . 6  |-  0  <_  1
3329, 30, 31, 32keephyp 3619 . . . . 5  |-  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1
3433a1i 10 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 )
35 dchrisum0flblem1.2 . . . . . . . . . 10  |-  ( ph  ->  A  e.  NN0 )
36 nn0uz 10262 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
3735, 36syl6eleq 2373 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( ZZ>= ` 
0 ) )
38 fzn0 10809 . . . . . . . . 9  |-  ( ( 0 ... A )  =/=  (/)  <->  A  e.  ( ZZ>=
`  0 ) )
3937, 38sylibr 203 . . . . . . . 8  |-  ( ph  ->  ( 0 ... A
)  =/=  (/) )
40 hashnncl 11354 . . . . . . . . 9  |-  ( ( 0 ... A )  e.  Fin  ->  (
( # `  ( 0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
417, 40syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
4239, 41mpbird 223 . . . . . . 7  |-  ( ph  ->  ( # `  (
0 ... A ) )  e.  NN )
4342adantr 451 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  NN )
4443nnge1d 9788 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  (
# `  ( 0 ... A ) ) )
45 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( X `  ( L `  P
) )  =  1 )
4645oveq1d 5873 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( X `  ( L `  P ) ) ^
i )  =  ( 1 ^ i ) )
47 elfzelz 10798 . . . . . . . . 9  |-  ( i  e.  ( 0 ... A )  ->  i  e.  ZZ )
48 1exp 11131 . . . . . . . . 9  |-  ( i  e.  ZZ  ->  (
1 ^ i )  =  1 )
4947, 48syl 15 . . . . . . . 8  |-  ( i  e.  ( 0 ... A )  ->  (
1 ^ i )  =  1 )
5046, 49sylan9eq 2335 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  =  1 )
5150sumeq2dv 12176 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  sum_ i  e.  ( 0 ... A ) 1 )
52 fzfid 11035 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( 0 ... A )  e. 
Fin )
53 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
54 fsumconst 12252 . . . . . . 7  |-  ( ( ( 0 ... A
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 0 ... A ) 1  =  ( (
# `  ( 0 ... A ) )  x.  1 ) )
5552, 53, 54sylancl 643 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) 1  =  ( ( # `  (
0 ... A ) )  x.  1 ) )
5643nncnd 9762 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  CC )
5756mulid1d 8852 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( # `
 ( 0 ... A ) )  x.  1 )  =  (
# `  ( 0 ... A ) ) )
5851, 55, 573eqtrd 2319 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  (
# `  ( 0 ... A ) ) )
5944, 58breqtrrd 4049 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
605, 6, 28, 34, 59letrd 8973 . . 3  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
61 oveq1 5865 . . . . . . 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 ) ) ) ) )
6261breq1d 4033 . . . . . 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 ) ) ) ) )
63 oveq1 5865 . . . . . . 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 ) ) ) ) )
6463breq1d 4033 . . . . . 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 ) ) ) ) )
6523adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  RR )
66 resubcl 9111 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( X `  ( L `
 P ) )  e.  RR )  -> 
( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
671, 65, 66sylancr 644 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
6867adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  RR )
6968leidd 9339 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  <_  ( 1  -  ( X `  ( L `  P )
) ) )
7067recnd 8861 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  CC )
7170adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  CC )
7271mulid2d 8853 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  =  ( 1  -  ( X `  ( L `  P ) ) ) )
73 nn0p1nn 10003 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
7435, 73syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN )
7574ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( A  +  1 )  e.  NN )
76750expd 11261 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( 0 ^ ( A  +  1 ) )  =  0 )
77 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( X `  ( L `  P )
)  =  0 )
7877oveq1d 5873 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( 0 ^ ( A  +  1 ) ) )
7976, 78, 773eqtr4d 2325 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
80 neg1cn 9813 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
8135ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  NN0 )
82 expp1 11110 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  A  e.  NN0 )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u 1 ^ A )  x.  -u 1
) )
8380, 81, 82sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u
1 ^ A )  x.  -u 1 ) )
84 prmnn 12761 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P  e.  Prime  ->  P  e.  NN )
8517, 84syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  P  e.  NN )
8685, 35nnexpcld 11266 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( P ^ A
)  e.  NN )
8786nncnd 9762 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( P ^ A
)  e.  CC )
8887ad2antrr 706 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P ^ A )  e.  CC )
8988sqsqrd 11921 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( sqr `  ( P ^ A ) ) ^ 2 )  =  ( P ^ A
) )
9089oveq2d 5874 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( P  pCnt  ( P ^ A ) ) )
9117ad2antrr 706 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  P  e.  Prime )
92 nnq 10329 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
9392adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
94 nnne0 9778 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
9594adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
96 2z 10054 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
9796a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  ZZ )
98 pcexp 12912 . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
9991, 93, 95, 97, 98syl121anc 1187 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( 2  x.  ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) ) )
10081nn0zd 10115 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  ZZ )
101 pcid 12925 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10291, 100, 101syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10390, 99, 1023eqtr3rd 2324 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  =  ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) )
104103oveq2d 5874 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  ( -u 1 ^ ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) ) )
10580a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  -u 1  e.  CC )
106 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  NN )
10791, 106pccld 12903 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  NN0 )
108 2nn0 9982 . . . . . . . . . . . . . . . . 17  |-  2  e.  NN0
109108a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  NN0 )
110105, 107, 109expmuld 11248 . . . . . . . . . . . . . . 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 ) ) ) ) )
111 sqneg 11164 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  e.  CC  ->  ( -u 1 ^ 2 )  =  ( 1 ^ 2 ) )
11253, 111ax-mp 8 . . . . . . . . . . . . . . . . . 18  |-  ( -u
1 ^ 2 )  =  ( 1 ^ 2 )
113 sq1 11198 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ^ 2 )  =  1
114112, 113eqtri 2303 . . . . . . . . . . . . . . . . 17  |-  ( -u
1 ^ 2 )  =  1
115114oveq1i 5868 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  ( 1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )
116107nn0zd 10115 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  ZZ )
117 1exp 11131 . . . . . . . . . . . . . . . . 17  |-  ( ( P  pCnt  ( sqr `  ( P ^ A
) ) )  e.  ZZ  ->  ( 1 ^ ( P  pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
118116, 117syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
119115, 118syl5eq 2327 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
120104, 110, 1193eqtrd 2319 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  1 )
121120oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
12280mulid2i 8840 . . . . . . . . . . . . 13  |-  ( 1  x.  -u 1 )  = 
-u 1
123121, 122syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  -u 1
)
12483, 123eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  -u 1 )
125124adantr 451 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( -u 1 ^ ( A  +  1 ) )  =  -u 1
)
12623recnd 8861 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  CC )
127126adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  CC )
128127ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  CC )
129128negnegd 9148 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  ( X `
 ( L `  P ) ) )
130 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  =/=  1
)
131130ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =/=  1 )
132 rpvmasum2.g . . . . . . . . . . . . . . . . . . 19  |-  G  =  (DChr `  N )
133 rpvmasum2.d . . . . . . . . . . . . . . . . . . 19  |-  D  =  ( Base `  G
)
134 dchrisum0f.x . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  X  e.  D )
135134ad3antrrr 710 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  X  e.  D )
136 eqid 2283 . . . . . . . . . . . . . . . . . . 19  |-  (Unit `  Z )  =  (Unit `  Z )
137132, 11, 133, 12, 136, 134, 21dchrn0 20489 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( X `  ( L `  P ) )  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
138137ad2antrr 706 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
)  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
139138biimpa 470 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( L `  P
)  e.  (Unit `  Z ) )
140132, 133, 135, 11, 136, 139dchrabs 20499 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  1 )
141 eqeq1 2289 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  ( X `
 ( L `  P ) ) )  =  ( X `  ( L `  P ) )  ->  ( ( abs `  ( X `  ( L `  P ) ) )  =  1  <-> 
( X `  ( L `  P )
)  =  1 ) )
142140, 141syl5ibcom 211 . . . . . . . . . . . . . . . . 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 ) )
143142necon3ad 2482 . . . . . . . . . . . . . . . 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 ) ) ) )
144131, 143mpd 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -.  ( abs `  ( X `  ( L `  P ) ) )  =  ( X `  ( L `  P ) ) )
14565ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  RR )
146145absord 11898 . . . . . . . . . . . . . . . 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 ) ) ) )
147146ord 366 . . . . . . . . . . . . . . 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 ) ) ) )
148144, 147mpd 14 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  -u ( X `  ( L `  P ) ) )
149148, 140eqtr3d 2317 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u ( X `  ( L `  P )
)  =  1 )
150149negeqd 9046 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  -u 1
)
151129, 150eqtr3d 2317 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =  -u 1
)
152151oveq1d 5873 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( -u 1 ^ ( A  + 
1 ) ) )
153125, 152, 1513eqtr4d 2325 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
15479, 153pm2.61dane 2524 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
155154oveq2d 5874 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  =  ( 1  -  ( X `  ( L `  P )
) ) )
15669, 72, 1553brtr4d 4053 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
15770mul02d 9010 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  =  0 )
158 peano2nn0 10004 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
15935, 158syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
16023, 159reexpcld 11262 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )
161160adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  RR )
162161recnd 8861 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  CC )
163162abscld 11918 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  e.  RR )
1641a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  e.  RR )
165161leabsd 11897 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  ( abs `  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
166159adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e. 
NN0 )
167127, 166absexpd 11934 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  =  ( ( abs `  ( X `
 ( L `  P ) ) ) ^ ( A  + 
1 ) ) )
168127abscld 11918 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  e.  RR )
169127absge0d 11926 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( abs `  ( X `
 ( L `  P ) ) ) )
170132, 133, 11, 12, 134, 21dchrabs2 20501 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( X `  ( L `  P ) ) )  <_  1 )
171170adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  <_  1
)
172 exple1 11161 . . . . . . . . . . . 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 )
173168, 169, 171, 166, 172syl31anc 1185 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( abs `  ( X `  ( L `  P ) ) ) ^ ( A  +  1 ) )  <_  1 )
174167, 173eqbrtrd 4043 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <_  1 )
175161, 163, 164, 165, 174letrd 8973 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  1
)
176 subge0 9287 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 0  <_  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  <-> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  <_  1 ) )
1771, 161, 176sylancr 644 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  <_  ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <->  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) )  <_  1
) )
178175, 177mpbird 223 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( 1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) ) )
179157, 178eqbrtrd 4043 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
180179adantr 451 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
( 0  x.  (
1  -  ( X `
 ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
18162, 64, 156, 180ifbothda 3595 . . . . 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 ) ) ) )
1821, 3keepel 3622 . . . . . . 7  |-  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR
183182a1i 10 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR )
184 resubcl 9111 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) )  e.  RR )
1851, 161, 184sylancr 644 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  e.  RR )
186130necomd 2529 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  =/=  ( X `  ( L `
 P ) ) )
18765leabsd 11897 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  ( abs `  ( X `  ( L `  P ) ) ) )
18865, 168, 164, 187, 171letrd 8973 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  1
)
18965, 164, 188leltned 8970 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  1  =/=  ( X `  ( L `  P ) ) ) )
190186, 189mpbird 223 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <  1
)
191 posdif 9267 . . . . . . . 8  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  1  e.  RR )  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  ( 1  -  ( X `  ( L `  P )
) ) ) )
19265, 1, 191sylancl 643 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  (
1  -  ( X `
 ( L `  P ) ) ) ) )
193190, 192mpbid 201 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <  ( 1  -  ( X `
 ( L `  P ) ) ) )
194 lemuldiv 9635 . . . . . 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 ) ) ) ) ) )
195183, 185, 67, 193, 194syl112anc 1186 . . . . 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 ) ) ) ) ) )
196181, 195mpbid 201 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
( ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  /  ( 1  -  ( X `  ( L `  P ) ) ) ) )
19735nn0zd 10115 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
198 fzval3 10911 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
199197, 198syl 15 . . . . . . 7  |-  ( ph  ->  ( 0 ... A
)  =  ( 0..^ ( A  +  1 ) ) )
200199adantr 451 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
201200sumeq1d 12174 . . . . 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 ) )
202 0nn0 9980 . . . . . . 7  |-  0  e.  NN0
203202a1i 10 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  e.  NN0 )
204159, 36syl6eleq 2373 . . . . . . 7  |-  ( ph  ->  ( A  +  1 )  e.  ( ZZ>= ` 
0 ) )
205204adantr 451 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e.  ( ZZ>= `  0 )
)
206127, 130, 203, 205geoserg 12324 . . . . 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 ) ) ) ) )
207127exp0d 11239 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
0 )  =  1 )
208207oveq1d 5873 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( (
( X `  ( L `  P )
) ^ 0 )  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  =  ( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
209208oveq1d 5873 . . . . 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 )
) ) ) )
210201, 206, 2093eqtrd 2319 . . . 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 ) ) ) ) )
211196, 210breqtrrd 4049 . . 3  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
21260, 211pm2.61dane 2524 . 2  |-  ( ph  ->  if ( ( sqr `  ( P ^ A
) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
213 rpvmasum2.1 . . . . 5  |-  .1.  =  ( 0g `  G )
214 dchrisum0f.f . . . . 5  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
21511, 13, 9, 132, 133, 213, 214dchrisum0fval 20654 . . . 4  |-  ( ( P ^ A )  e.  NN  ->  ( F `  ( P ^ A ) )  = 
sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
21686, 215syl 15 . . 3  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
217 fveq2 5525 . . . . 5  |-  ( k  =  ( P ^
i )  ->  ( L `  k )  =  ( L `  ( P ^ i ) ) )
218217fveq2d 5529 . . . 4  |-  ( k  =  ( P ^
i )  ->  ( X `  ( L `  k ) )  =  ( X `  ( L `  ( P ^ i ) ) ) )
219 eqid 2283 . . . . . 6  |-  ( b  e.  ( 0 ... A )  |->  ( P ^ b ) )  =  ( b  e.  ( 0 ... A
)  |->  ( P ^
b ) )
220219dvdsppwf1o 20426 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
b  e.  ( 0 ... A )  |->  ( P ^ b ) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  |  q 
||  ( P ^ A ) } )
22117, 35, 220syl2anc 642 . . . 4  |-  ( ph  ->  ( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  | 
q  ||  ( P ^ A ) } )
222 oveq2 5866 . . . . . 6  |-  ( b  =  i  ->  ( P ^ b )  =  ( P ^ i
) )
223 ovex 5883 . . . . . 6  |-  ( P ^ b )  e. 
_V
224222, 219, 223fvmpt3i 5605 . . . . 5  |-  ( i  e.  ( 0 ... A )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
225224adantl 452 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
2268adantr 451 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  X : (
Base `  Z ) --> RR )
227 ssrab2 3258 . . . . . . . . 9  |-  { q  e.  NN  |  q 
||  ( P ^ A ) }  C_  NN
228227sseli 3176 . . . . . . . 8  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  NN )
229228nnzd 10116 . . . . . . 7  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  ZZ )
230 ffvelrn 5663 . . . . . . 7  |-  ( ( L : ZZ --> ( Base `  Z )  /\  k  e.  ZZ )  ->  ( L `  k )  e.  ( Base `  Z
) )
23116, 229, 230syl2an 463 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( L `  k )  e.  (
Base `  Z )
)
232 ffvelrn 5663 . . . . . 6  |-  ( ( X : ( Base `  Z ) --> RR  /\  ( L `  k )  e.  ( Base `  Z
) )  ->  ( X `  ( L `  k ) )  e.  RR )
233226, 231, 232syl2anc 642 . . . . 5  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  RR )
234233recnd 8861 . . . 4  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  CC )
235218, 7, 221, 225, 234fsumf1o 12196 . . 3  |-  ( ph  -> 
sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) )  = 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) ) )
236 zsubrg 16425 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
237 eqid 2283 . . . . . . . . . . . 12  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
238237subrgsubm 15558 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
239236, 238mp1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
24024adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  i  e.  NN0 )
24119adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  P  e.  ZZ )
242 eqid 2283 . . . . . . . . . . 11  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
243 cnfldex 16380 . . . . . . . . . . . . 13  |-fld  e.  _V
244 zex 10033 . . . . . . . . . . . . 13  |-  ZZ  e.  _V
245 eqid 2283 . . . . . . . . . . . . . 14  |-  (flds  ZZ )  =  (flds  ZZ )
246245, 237mgpress 15336 . . . . . . . . . . . . 13  |-  ( (fld  e. 
_V  /\  ZZ  e.  _V )  ->  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp `  (flds  ZZ ) ) )
247243, 244, 246mp2an 653 . . . . . . . . . . . 12  |-  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp `  (flds  ZZ ) )
248247eqcomi 2287 . . . . . . . . . . 11  |-  (mulGrp `  (flds  ZZ ) )  =  ( (mulGrp ` fld )s  ZZ )
249 eqid 2283 . . . . . . . . . . 11  |-  (.g `  (mulGrp `  (flds  ZZ ) ) )  =  (.g `  (mulGrp `  (flds  ZZ )
) )
250242, 248, 249submmulg 14602 . . . . . . . . . 10  |-  ( ( ZZ  e.  (SubMnd `  (mulGrp ` fld ) )  /\  i  e.  NN0  /\  P  e.  ZZ )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp `  (flds  ZZ ) ) ) P ) )
251239, 240, 241, 250syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp `  (flds  ZZ ) ) ) P ) )
25285nncnd 9762 . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
253 cnfldexp 16407 . . . . . . . . . 10  |-  ( ( P  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^ i
) )
254252, 24, 253syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^
i ) )
255251, 254eqtr3d 2317 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp `  (flds  ZZ )
) ) P )  =  ( P ^
i ) )
256255fveq2d 5529 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp `  (flds  ZZ )
) ) P ) )  =  ( L `
 ( P ^
i ) ) )
25711zncrng 16498 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
258 crngrng 15351 . . . . . . . . . . 11  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
25910, 257, 2583syl 18 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  Ring )
260245, 13zrhrhm 16466 . . . . . . . . . 10  |-  ( Z  e.  Ring  ->  L  e.  ( (flds  ZZ ) RingHom  Z ) )
261 eqid 2283 . . . . . . . . . . 11  |-  (mulGrp `  (flds  ZZ ) )  =  (mulGrp `  (flds  ZZ ) )
262 eqid 2283 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
263261, 262rhmmhm 15502 . . . . . . . . . 10  |-  ( L  e.  ( (flds  ZZ ) RingHom  Z )  ->  L  e.  ( (mulGrp `  (flds  ZZ ) ) MndHom  (mulGrp `  Z ) ) )
264259, 260, 2633syl 18 . . . . . . . . 9  |-  ( ph  ->  L  e.  ( (mulGrp `  (flds  ZZ ) ) MndHom  (mulGrp `  Z ) ) )
265264adantr 451 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  L  e.  ( (mulGrp `  (flds  ZZ )
) MndHom  (mulGrp `  Z )
) )
266 subrgsubg 15551 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
267245subgbas 14625 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubGrp ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
268236, 266, 267mp2b 9 . . . . . . . . . 10  |-  ZZ  =  ( Base `  (flds  ZZ ) )
269261, 268mgpbas 15331 . . . . . . . . 9  |-  ZZ  =  ( Base `  (mulGrp `  (flds  ZZ )
) )
270 eqid 2283 . . . . . . . . 9  |-  (.g `  (mulGrp `  Z ) )  =  (.g `  (mulGrp `  Z
) )
271269, 249, 270mhmmulg 14599 . . . . . . . 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 ) ) )
272265, 240, 241, 271syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp `  (flds  ZZ )
) ) P ) )  =  ( i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )
273256, 272eqtr3d 2317 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( P ^ i ) )  =  ( i (.g `  (mulGrp `  Z )
) ( L `  P ) ) )
274273fveq2d 5529 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( X `  (
i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) ) )
275132, 11, 133dchrmhm 20480 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
276275, 134sseldi 3178 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
277276adantr 451 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
27821adantr 451 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  P )  e.  ( Base `  Z
) )
279262, 12mgpbas 15331 . . . . . . 7  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
280279, 270, 242mhmmulg 14599 . . . . . 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 ) ) ) )
281277, 240, 278, 280syl3anc 1182 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( i
(.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )  =  ( i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) ) )
282 cnfldexp 16407 . . . . . 6  |-  ( ( ( X `  ( L `  P )
)  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) ( X `  ( L `  P ) ) )  =  ( ( X `  ( L `  P )
) ^ i ) )
283126, 24, 282syl2an 463 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) )  =  ( ( X `
 ( L `  P ) ) ^
i ) )
284274, 281, 2833eqtrd 2319 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( ( X `  ( L `  P ) ) ^ i ) )
285284sumeq2dv 12176 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) )  =  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
286216, 235, 2853eqtrd 2319 . 2  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i ) )
287212, 286breqtrrd 4049 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788   (/)c0 3455   ifcif 3565   class class class wbr 4023    e. cmpt 4077   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   QQcq 10316   ...cfz 10782  ..^cfzo 10870   ^cexp 11104   #chash 11337   sqrcsqr 11718   abscabs 11719   sum_csu 12158    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   ↾s cress 13149   0gc0g 13400  .gcmg 14366   MndHom cmhm 14413  SubMndcsubmnd 14414  SubGrpcsubg 14615  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338  Unitcui 15421   RingHom crh 15494  SubRingcsubrg 15541  ℂfldccnfld 16377   ZRHomczrh 16451  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrisum0flblem2  20658  dchrisum0flb  20659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-od 14844  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-dchr 20472
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