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Theorem dchrisum0flblem1 22737
Description: Lemma for dchrisum0flb 22739. 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 1red 9393 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  1  e.  RR )
2 0red 9379 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
0  e.  RR )
31, 2ifclda 3816 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR )
4 1red 9393 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  e.  RR )
5 fzfid 11787 . . . . . 6  |-  ( ph  ->  ( 0 ... A
)  e.  Fin )
6 dchrisum0flb.r . . . . . . . 8  |-  ( ph  ->  X : ( Base `  Z ) --> RR )
7 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
87nnnn0d 10628 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
9 rpvmasum.z . . . . . . . . . . 11  |-  Z  =  (ℤ/n `  N )
10 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  Z )  =  (
Base `  Z )
11 rpvmasum.l . . . . . . . . . . 11  |-  L  =  ( ZRHom `  Z
)
129, 10, 11znzrhfo 17960 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
13 fof 5615 . . . . . . . . . 10  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
148, 12, 133syl 20 . . . . . . . . 9  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
15 dchrisum0flblem1.1 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Prime )
16 prmz 13759 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1715, 16syl 16 . . . . . . . . 9  |-  ( ph  ->  P  e.  ZZ )
1814, 17ffvelrnd 5839 . . . . . . . 8  |-  ( ph  ->  ( L `  P
)  e.  ( Base `  Z ) )
196, 18ffvelrnd 5839 . . . . . . 7  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  RR )
20 elfznn0 11473 . . . . . . 7  |-  ( i  e.  ( 0 ... A )  ->  i  e.  NN0 )
21 reexpcl 11874 . . . . . . 7  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  i  e.  NN0 )  -> 
( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2219, 20, 21syl2an 477 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  e.  RR )
235, 22fsumrecl 13203 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2423adantr 465 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  e.  RR )
25 breq1 4290 . . . . . 6  |-  ( 1  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 1  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
26 breq1 4290 . . . . . 6  |-  ( 0  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 0  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
27 1le1 9956 . . . . . 6  |-  1  <_  1
28 0le1 9855 . . . . . 6  |-  0  <_  1
2925, 26, 27, 28keephyp 3849 . . . . 5  |-  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1
3029a1i 11 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 )
31 dchrisum0flblem1.2 . . . . . . . . . 10  |-  ( ph  ->  A  e.  NN0 )
32 nn0uz 10887 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
3331, 32syl6eleq 2528 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( ZZ>= ` 
0 ) )
34 fzn0 11456 . . . . . . . . 9  |-  ( ( 0 ... A )  =/=  (/)  <->  A  e.  ( ZZ>=
`  0 ) )
3533, 34sylibr 212 . . . . . . . 8  |-  ( ph  ->  ( 0 ... A
)  =/=  (/) )
36 hashnncl 12126 . . . . . . . . 9  |-  ( ( 0 ... A )  e.  Fin  ->  (
( # `  ( 0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
375, 36syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
3835, 37mpbird 232 . . . . . . 7  |-  ( ph  ->  ( # `  (
0 ... A ) )  e.  NN )
3938adantr 465 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  NN )
4039nnge1d 10356 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  (
# `  ( 0 ... A ) ) )
41 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( X `  ( L `  P
) )  =  1 )
4241oveq1d 6101 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( X `  ( L `  P ) ) ^
i )  =  ( 1 ^ i ) )
43 elfzelz 11445 . . . . . . . . 9  |-  ( i  e.  ( 0 ... A )  ->  i  e.  ZZ )
44 1exp 11885 . . . . . . . . 9  |-  ( i  e.  ZZ  ->  (
1 ^ i )  =  1 )
4543, 44syl 16 . . . . . . . 8  |-  ( i  e.  ( 0 ... A )  ->  (
1 ^ i )  =  1 )
4642, 45sylan9eq 2490 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  =  1 )
4746sumeq2dv 13172 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  sum_ i  e.  ( 0 ... A ) 1 )
48 fzfid 11787 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( 0 ... A )  e. 
Fin )
49 ax-1cn 9332 . . . . . . 7  |-  1  e.  CC
50 fsumconst 13249 . . . . . . 7  |-  ( ( ( 0 ... A
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 0 ... A ) 1  =  ( (
# `  ( 0 ... A ) )  x.  1 ) )
5148, 49, 50sylancl 662 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) 1  =  ( ( # `  (
0 ... A ) )  x.  1 ) )
5239nncnd 10330 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  CC )
5352mulid1d 9395 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( # `
 ( 0 ... A ) )  x.  1 )  =  (
# `  ( 0 ... A ) ) )
5447, 51, 533eqtrd 2474 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  (
# `  ( 0 ... A ) ) )
5540, 54breqtrrd 4313 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
563, 4, 24, 30, 55letrd 9520 . . 3  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
57 oveq1 6093 . . . . . . 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 ) ) ) ) )
5857breq1d 4297 . . . . . 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 ) ) ) ) )
59 oveq1 6093 . . . . . . 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 ) ) ) ) )
6059breq1d 4297 . . . . . 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 ) ) ) ) )
61 1re 9377 . . . . . . . . . 10  |-  1  e.  RR
6219adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  RR )
63 resubcl 9665 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( X `  ( L `
 P ) )  e.  RR )  -> 
( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
6461, 62, 63sylancr 663 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
6564adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  RR )
6665leidd 9898 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  <_  ( 1  -  ( X `  ( L `  P )
) ) )
6764recnd 9404 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  CC )
6867adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  CC )
6968mulid2d 9396 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  =  ( 1  -  ( X `  ( L `  P ) ) ) )
70 nn0p1nn 10611 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
7131, 70syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN )
7271ad3antrrr 729 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( A  +  1 )  e.  NN )
73720expd 12016 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( 0 ^ ( A  +  1 ) )  =  0 )
74 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( X `  ( L `  P )
)  =  0 )
7574oveq1d 6101 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( 0 ^ ( A  +  1 ) ) )
7673, 75, 743eqtr4d 2480 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
77 neg1cn 10417 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
7831ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  NN0 )
79 expp1 11864 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  A  e.  NN0 )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u 1 ^ A )  x.  -u 1
) )
8077, 78, 79sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u
1 ^ A )  x.  -u 1 ) )
81 prmnn 13758 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P  e.  Prime  ->  P  e.  NN )
8215, 81syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  P  e.  NN )
8382, 31nnexpcld 12021 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( P ^ A
)  e.  NN )
8483nncnd 10330 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( P ^ A
)  e.  CC )
8584ad2antrr 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P ^ A )  e.  CC )
8685sqsqrd 12917 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( sqr `  ( P ^ A ) ) ^ 2 )  =  ( P ^ A
) )
8786oveq2d 6102 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( P  pCnt  ( P ^ A ) ) )
8815ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  P  e.  Prime )
89 nnq 10958 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
9089adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
91 nnne0 10346 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
9291adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
93 2z 10670 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
9493a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  ZZ )
95 pcexp 13918 . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
9688, 90, 92, 94, 95syl121anc 1223 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( 2  x.  ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) ) )
9778nn0zd 10737 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  ZZ )
98 pcid 13931 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
9988, 97, 98syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10087, 96, 993eqtr3rd 2479 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  =  ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) )
101100oveq2d 6102 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  ( -u 1 ^ ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) ) )
10277a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  -u 1  e.  CC )
103 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  NN )
10488, 103pccld 13909 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  NN0 )
105 2nn0 10588 . . . . . . . . . . . . . . . . 17  |-  2  e.  NN0
106105a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  NN0 )
107102, 104, 106expmuld 12003 . . . . . . . . . . . . . . 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 ) ) ) ) )
108 neg1sqe1 11953 . . . . . . . . . . . . . . . . 17  |-  ( -u
1 ^ 2 )  =  1
109108oveq1i 6096 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  ( 1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )
110104nn0zd 10737 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  ZZ )
111 1exp 11885 . . . . . . . . . . . . . . . . 17  |-  ( ( P  pCnt  ( sqr `  ( P ^ A
) ) )  e.  ZZ  ->  ( 1 ^ ( P  pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
112110, 111syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
113109, 112syl5eq 2482 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
114101, 107, 1133eqtrd 2474 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  1 )
115114oveq1d 6101 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
11677mulid2i 9381 . . . . . . . . . . . . 13  |-  ( 1  x.  -u 1 )  = 
-u 1
117115, 116syl6eq 2486 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  -u 1
)
11880, 117eqtrd 2470 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  -u 1 )
119118adantr 465 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( -u 1 ^ ( A  +  1 ) )  =  -u 1
)
12019recnd 9404 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  CC )
121120adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  CC )
122121ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  CC )
123122negnegd 9702 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  ( X `
 ( L `  P ) ) )
124 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  =/=  1
)
125124ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =/=  1 )
126 rpvmasum2.g . . . . . . . . . . . . . . . . . . 19  |-  G  =  (DChr `  N )
127 rpvmasum2.d . . . . . . . . . . . . . . . . . . 19  |-  D  =  ( Base `  G
)
128 dchrisum0f.x . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  X  e.  D )
129128ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  X  e.  D )
130 eqid 2438 . . . . . . . . . . . . . . . . . . 19  |-  (Unit `  Z )  =  (Unit `  Z )
131126, 9, 127, 10, 130, 128, 18dchrn0 22569 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( X `  ( L `  P ) )  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
132131ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
)  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
133132biimpa 484 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( L `  P
)  e.  (Unit `  Z ) )
134126, 127, 129, 9, 130, 133dchrabs 22579 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  1 )
135 eqeq1 2444 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  ( X `
 ( L `  P ) ) )  =  ( X `  ( L `  P ) )  ->  ( ( abs `  ( X `  ( L `  P ) ) )  =  1  <-> 
( X `  ( L `  P )
)  =  1 ) )
136134, 135syl5ibcom 220 . . . . . . . . . . . . . . . . 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 ) )
137136necon3ad 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 ) ) ) )
138125, 137mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -.  ( abs `  ( X `  ( L `  P ) ) )  =  ( X `  ( L `  P ) ) )
13962ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  RR )
140139absord 12894 . . . . . . . . . . . . . . . 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 ) ) ) )
141140ord 377 . . . . . . . . . . . . . . 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 ) ) ) )
142138, 141mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  -u ( X `  ( L `  P ) ) )
143142, 134eqtr3d 2472 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u ( X `  ( L `  P )
)  =  1 )
144143negeqd 9596 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  -u 1
)
145123, 144eqtr3d 2472 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =  -u 1
)
146145oveq1d 6101 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( -u 1 ^ ( A  + 
1 ) ) )
147119, 146, 1453eqtr4d 2480 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
14876, 147pm2.61dane 2684 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
149148oveq2d 6102 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  =  ( 1  -  ( X `  ( L `  P )
) ) )
15066, 69, 1493brtr4d 4317 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
15167mul02d 9559 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  =  0 )
152 peano2nn0 10612 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
15331, 152syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
15419, 153reexpcld 12017 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )
155154adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  RR )
156155recnd 9404 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  CC )
157156abscld 12914 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  e.  RR )
158 1red 9393 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  e.  RR )
159155leabsd 12893 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  ( abs `  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
160153adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e. 
NN0 )
161121, 160absexpd 12930 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  =  ( ( abs `  ( X `
 ( L `  P ) ) ) ^ ( A  + 
1 ) ) )
162121abscld 12914 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  e.  RR )
163121absge0d 12922 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( abs `  ( X `
 ( L `  P ) ) ) )
164126, 127, 9, 10, 128, 18dchrabs2 22581 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( X `  ( L `  P ) ) )  <_  1 )
165164adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  <_  1
)
166 exple1 11915 . . . . . . . . . . . 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 )
167162, 163, 165, 160, 166syl31anc 1221 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( abs `  ( X `  ( L `  P ) ) ) ^ ( A  +  1 ) )  <_  1 )
168161, 167eqbrtrd 4307 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <_  1 )
169155, 157, 158, 159, 168letrd 9520 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  1
)
170 subge0 9844 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 0  <_  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  <-> 
( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  <_  1 ) )
17161, 155, 170sylancr 663 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  <_  ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <->  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) )  <_  1
) )
172169, 171mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( 1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) ) )
173151, 172eqbrtrd 4307 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
174173adantr 465 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
( 0  x.  (
1  -  ( X `
 ( L `  P ) ) ) )  <_  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
17558, 60, 150, 174ifbothda 3819 . . . . 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 ) ) ) )
176 0re 9378 . . . . . . . 8  |-  0  e.  RR
17761, 176keepel 3852 . . . . . . 7  |-  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR
178177a1i 11 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR )
179 resubcl 9665 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) )  e.  RR )
18061, 155, 179sylancr 663 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  e.  RR )
181124necomd 2690 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  =/=  ( X `  ( L `
 P ) ) )
18262leabsd 12893 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  ( abs `  ( X `  ( L `  P ) ) ) )
18362, 162, 158, 182, 165letrd 9520 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  1
)
18462, 158, 183leltned 9517 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  1  =/=  ( X `  ( L `  P ) ) ) )
185181, 184mpbird 232 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <  1
)
186 posdif 9824 . . . . . . . 8  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  1  e.  RR )  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  ( 1  -  ( X `  ( L `  P )
) ) ) )
18762, 61, 186sylancl 662 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  (
1  -  ( X `
 ( L `  P ) ) ) ) )
188185, 187mpbid 210 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <  ( 1  -  ( X `
 ( L `  P ) ) ) )
189 lemuldiv 10203 . . . . . 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 ) ) ) ) ) )
190178, 180, 64, 188, 189syl112anc 1222 . . . . 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 ) ) ) ) ) )
191175, 190mpbid 210 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
( ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  /  ( 1  -  ( X `  ( L `  P ) ) ) ) )
19231nn0zd 10737 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
193 fzval3 11597 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
194192, 193syl 16 . . . . . . 7  |-  ( ph  ->  ( 0 ... A
)  =  ( 0..^ ( A  +  1 ) ) )
195194adantr 465 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
196195sumeq1d 13170 . . . . 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 ) )
197 0nn0 10586 . . . . . . 7  |-  0  e.  NN0
198197a1i 11 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  e.  NN0 )
199153, 32syl6eleq 2528 . . . . . . 7  |-  ( ph  ->  ( A  +  1 )  e.  ( ZZ>= ` 
0 ) )
200199adantr 465 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e.  ( ZZ>= `  0 )
)
201121, 124, 198, 200geoserg 13320 . . . . 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 ) ) ) ) )
202121exp0d 11994 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
0 )  =  1 )
203202oveq1d 6101 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( (
( X `  ( L `  P )
) ^ 0 )  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  =  ( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
204203oveq1d 6101 . . . . 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 )
) ) ) )
205196, 201, 2043eqtrd 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 ) ) ) ) )
206191, 205breqtrrd 4313 . . 3  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
20756, 206pm2.61dane 2684 . 2  |-  ( ph  ->  if ( ( sqr `  ( P ^ A
) )  e.  NN ,  1 ,  0 )  <_  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
208 rpvmasum2.1 . . . . 5  |-  .1.  =  ( 0g `  G )
209 dchrisum0f.f . . . . 5  |-  F  =  ( b  e.  NN  |->  sum_ v  e.  { q  e.  NN  |  q 
||  b }  ( X `  ( L `  v ) ) )
2109, 11, 7, 126, 127, 208, 209dchrisum0fval 22734 . . . 4  |-  ( ( P ^ A )  e.  NN  ->  ( F `  ( P ^ A ) )  = 
sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
21183, 210syl 16 . . 3  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
212 fveq2 5686 . . . . 5  |-  ( k  =  ( P ^
i )  ->  ( L `  k )  =  ( L `  ( P ^ i ) ) )
213212fveq2d 5690 . . . 4  |-  ( k  =  ( P ^
i )  ->  ( X `  ( L `  k ) )  =  ( X `  ( L `  ( P ^ i ) ) ) )
214 eqid 2438 . . . . . 6  |-  ( b  e.  ( 0 ... A )  |->  ( P ^ b ) )  =  ( b  e.  ( 0 ... A
)  |->  ( P ^
b ) )
215214dvdsppwf1o 22506 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
b  e.  ( 0 ... A )  |->  ( P ^ b ) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  |  q 
||  ( P ^ A ) } )
21615, 31, 215syl2anc 661 . . . 4  |-  ( ph  ->  ( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  | 
q  ||  ( P ^ A ) } )
217 oveq2 6094 . . . . . 6  |-  ( b  =  i  ->  ( P ^ b )  =  ( P ^ i
) )
218 ovex 6111 . . . . . 6  |-  ( P ^ b )  e. 
_V
219217, 214, 218fvmpt3i 5773 . . . . 5  |-  ( i  e.  ( 0 ... A )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
220219adantl 466 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
2216adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  X : (
Base `  Z ) --> RR )
222 elrabi 3109 . . . . . . . 8  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  NN )
223222nnzd 10738 . . . . . . 7  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  ZZ )
224 ffvelrn 5836 . . . . . . 7  |-  ( ( L : ZZ --> ( Base `  Z )  /\  k  e.  ZZ )  ->  ( L `  k )  e.  ( Base `  Z
) )
22514, 223, 224syl2an 477 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( L `  k )  e.  (
Base `  Z )
)
226221, 225ffvelrnd 5839 . . . . 5  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  RR )
227226recnd 9404 . . . 4  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  CC )
228213, 5, 216, 220, 227fsumf1o 13192 . . 3  |-  ( ph  -> 
sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) )  = 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) ) )
229 zsubrg 17846 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
230 eqid 2438 . . . . . . . . . . . 12  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
231230subrgsubm 16858 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
232229, 231mp1i 12 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
23320adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  i  e.  NN0 )
23417adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  P  e.  ZZ )
235 eqid 2438 . . . . . . . . . . 11  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
236 zringmpg 17896 . . . . . . . . . . . 12  |-  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp ` ring )
237236eqcomi 2442 . . . . . . . . . . 11  |-  (mulGrp ` ring )  =  ( (mulGrp ` fld )s  ZZ )
238 eqid 2438 . . . . . . . . . . 11  |-  (.g `  (mulGrp ` ring ) )  =  (.g `  (mulGrp ` ring ) )
239235, 237, 238submmulg 15653 . . . . . . . . . 10  |-  ( ( ZZ  e.  (SubMnd `  (mulGrp ` fld ) )  /\  i  e.  NN0  /\  P  e.  ZZ )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp ` ring ) ) P ) )
240232, 233, 234, 239syl3anc 1218 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp ` ring ) ) P ) )
24182nncnd 10330 . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
242 cnfldexp 17829 . . . . . . . . . 10  |-  ( ( P  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^ i
) )
243241, 20, 242syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^
i ) )
244240, 243eqtr3d 2472 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` ring ) ) P )  =  ( P ^
i ) )
245244fveq2d 5690 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp ` ring ) ) P ) )  =  ( L `
 ( P ^
i ) ) )
2469zncrng 17957 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
247 crngrng 16645 . . . . . . . . . . 11  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
2488, 246, 2473syl 20 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  Ring )
24911zrhrhm 17923 . . . . . . . . . 10  |-  ( Z  e.  Ring  ->  L  e.  (ring RingHom  Z ) )
250 eqid 2438 . . . . . . . . . . 11  |-  (mulGrp ` ring )  =  (mulGrp ` ring )
251 eqid 2438 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
252250, 251rhmmhm 16802 . . . . . . . . . 10  |-  ( L  e.  (ring RingHom  Z )  ->  L  e.  ( (mulGrp ` ring ) MndHom  (mulGrp `  Z
) ) )
253248, 249, 2523syl 20 . . . . . . . . 9  |-  ( ph  ->  L  e.  ( (mulGrp ` ring ) MndHom  (mulGrp `  Z )
) )
254253adantr 465 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  L  e.  ( (mulGrp ` ring ) MndHom  (mulGrp `  Z
) ) )
255 zringbas 17869 . . . . . . . . . 10  |-  ZZ  =  ( Base ` ring )
256250, 255mgpbas 16587 . . . . . . . . 9  |-  ZZ  =  ( Base `  (mulGrp ` ring ) )
257 eqid 2438 . . . . . . . . 9  |-  (.g `  (mulGrp `  Z ) )  =  (.g `  (mulGrp `  Z
) )
258256, 238, 257mhmmulg 15650 . . . . . . . 8  |-  ( ( L  e.  ( (mulGrp ` ring ) MndHom  (mulGrp `  Z )
)  /\  i  e.  NN0 
/\  P  e.  ZZ )  ->  ( L `  ( i (.g `  (mulGrp ` ring ) ) P ) )  =  ( i (.g `  (mulGrp `  Z )
) ( L `  P ) ) )
259254, 233, 234, 258syl3anc 1218 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp ` ring ) ) P ) )  =  ( i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )
260245, 259eqtr3d 2472 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( P ^ i ) )  =  ( i (.g `  (mulGrp `  Z )
) ( L `  P ) ) )
261260fveq2d 5690 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( X `  (
i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) ) )
262126, 9, 127dchrmhm 22560 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
263262, 128sseldi 3349 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
264263adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
26518adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  P )  e.  ( Base `  Z
) )
266251, 10mgpbas 16587 . . . . . . 7  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
267266, 257, 235mhmmulg 15650 . . . . . 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 ) ) ) )
268264, 233, 265, 267syl3anc 1218 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( i
(.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )  =  ( i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) ) )
269 cnfldexp 17829 . . . . . 6  |-  ( ( ( X `  ( L `  P )
)  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) ( X `  ( L `  P ) ) )  =  ( ( X `  ( L `  P )
) ^ i ) )
270120, 20, 269syl2an 477 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) )  =  ( ( X `
 ( L `  P ) ) ^
i ) )
271261, 268, 2703eqtrd 2474 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( ( X `  ( L `  P ) ) ^ i ) )
272271sumeq2dv 13172 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) )  =  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
273211, 228, 2723eqtrd 2474 . 2  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i ) )
274207, 273breqtrrd 4313 1  |-  ( ph  ->  if ( ( sqr `  ( P ^ A
) )  e.  NN ,  1 ,  0 )  <_  ( F `  ( P ^ A
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   {crab 2714   (/)c0 3632   ifcif 3786   class class class wbr 4287    e. cmpt 4345   -->wf 5409   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587   -ucneg 9588    / cdiv 9985   NNcn 10314   2c2 10363   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   QQcq 10945   ...cfz 11429  ..^cfzo 11540   ^cexp 11857   #chash 12095   sqrcsqr 12714   abscabs 12715   sum_csu 13155    || cdivides 13527   Primecprime 13755    pCnt cpc 13895   Basecbs 14166   ↾s cress 14167   0gc0g 14370  .gcmg 15406   MndHom cmhm 15454  SubMndcsubmnd 15455  mulGrpcmgp 16581   Ringcrg 16635   CRingccrg 16636  Unitcui 16721   RingHom crh 16794  SubRingcsubrg 16841  ℂfldccnfld 17798  ℤringzring 17863   ZRHomczrh 17911  ℤ/nczn 17914  DChrcdchr 22551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-divs 14439  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-nsg 15670  df-eqg 15671  df-ghm 15736  df-cntz 15826  df-od 16023  df-cmn 16270  df-abl 16271  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796  df-drng 16814  df-subrg 16843  df-lmod 16930  df-lss 16994  df-lsp 17033  df-sra 17233  df-rgmod 17234  df-lidl 17235  df-rsp 17236  df-2idl 17294  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-zring 17864  df-zrh 17915  df-zn 17918  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-log 21988  df-cxp 21989  df-dchr 22552
This theorem is referenced by:  dchrisum0flblem2  22738  dchrisum0flb  22739
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