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Theorem dchrisum0flblem1 24288
Description: Lemma for dchrisum0flb 24290. 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 9609 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  1  e.  RR )
2 0red 9595 . . . . 5  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  -.  ( sqr `  ( P ^ A ) )  e.  NN )  -> 
0  e.  RR )
31, 2ifclda 3886 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  if (
( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  e.  RR )
4 1red 9609 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  e.  RR )
5 fzfid 12136 . . . . . 6  |-  ( ph  ->  ( 0 ... A
)  e.  Fin )
6 dchrisum0flb.r . . . . . . . 8  |-  ( ph  ->  X : ( Base `  Z ) --> RR )
7 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
87nnnn0d 10876 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
9 rpvmasum.z . . . . . . . . . . 11  |-  Z  =  (ℤ/n `  N )
10 eqid 2428 . . . . . . . . . . 11  |-  ( Base `  Z )  =  (
Base `  Z )
11 rpvmasum.l . . . . . . . . . . 11  |-  L  =  ( ZRHom `  Z
)
129, 10, 11znzrhfo 19060 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
13 fof 5753 . . . . . . . . . 10  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
148, 12, 133syl 18 . . . . . . . . 9  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
15 dchrisum0flblem1.1 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Prime )
16 prmz 14569 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
1715, 16syl 17 . . . . . . . . 9  |-  ( ph  ->  P  e.  ZZ )
1814, 17ffvelrnd 5982 . . . . . . . 8  |-  ( ph  ->  ( L `  P
)  e.  ( Base `  Z ) )
196, 18ffvelrnd 5982 . . . . . . 7  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  RR )
20 elfznn0 11838 . . . . . . 7  |-  ( i  e.  ( 0 ... A )  ->  i  e.  NN0 )
21 reexpcl 12239 . . . . . . 7  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  i  e.  NN0 )  -> 
( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2219, 20, 21syl2an 479 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  e.  RR )
235, 22fsumrecl 13743 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i )  e.  RR )
2423adantr 466 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  e.  RR )
25 breq1 4369 . . . . . 6  |-  ( 1  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 1  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
26 breq1 4369 . . . . . 6  |-  ( 0  =  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  -> 
( 0  <_  1  <->  if ( ( sqr `  ( P ^ A ) )  e.  NN ,  1 ,  0 )  <_ 
1 ) )
27 1le1 10191 . . . . . 6  |-  1  <_  1
28 0le1 10088 . . . . . 6  |-  0  <_  1
2925, 26, 27, 28keephyp 3918 . . . . 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 11144 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
3331, 32syl6eleq 2516 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( ZZ>= ` 
0 ) )
34 fzn0 11764 . . . . . . . . 9  |-  ( ( 0 ... A )  =/=  (/)  <->  A  e.  ( ZZ>=
`  0 ) )
3533, 34sylibr 215 . . . . . . . 8  |-  ( ph  ->  ( 0 ... A
)  =/=  (/) )
36 hashnncl 12497 . . . . . . . . 9  |-  ( ( 0 ... A )  e.  Fin  ->  (
( # `  ( 0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
375, 36syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (
0 ... A ) )  e.  NN  <->  ( 0 ... A )  =/=  (/) ) )
3835, 37mpbird 235 . . . . . . 7  |-  ( ph  ->  ( # `  (
0 ... A ) )  e.  NN )
3938adantr 466 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  NN )
4039nnge1d 10603 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  (
# `  ( 0 ... A ) ) )
41 simpr 462 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( X `  ( L `  P
) )  =  1 )
4241oveq1d 6264 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( X `  ( L `  P ) ) ^
i )  =  ( 1 ^ i ) )
43 elfzelz 11751 . . . . . . . . 9  |-  ( i  e.  ( 0 ... A )  ->  i  e.  ZZ )
44 1exp 12251 . . . . . . . . 9  |-  ( i  e.  ZZ  ->  (
1 ^ i )  =  1 )
4543, 44syl 17 . . . . . . . 8  |-  ( i  e.  ( 0 ... A )  ->  (
1 ^ i )  =  1 )
4642, 45sylan9eq 2482 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =  1 )  /\  i  e.  ( 0 ... A
) )  ->  (
( X `  ( L `  P )
) ^ i )  =  1 )
4746sumeq2dv 13712 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  sum_ i  e.  ( 0 ... A ) 1 )
48 fzfid 12136 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( 0 ... A )  e. 
Fin )
49 ax-1cn 9548 . . . . . . 7  |-  1  e.  CC
50 fsumconst 13794 . . . . . . 7  |-  ( ( ( 0 ... A
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ i  e.  ( 0 ... A ) 1  =  ( (
# `  ( 0 ... A ) )  x.  1 ) )
5148, 49, 50sylancl 666 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) 1  =  ( ( # `  (
0 ... A ) )  x.  1 ) )
5239nncnd 10576 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( # `  (
0 ... A ) )  e.  CC )
5352mulid1d 9611 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  ( ( # `
 ( 0 ... A ) )  x.  1 )  =  (
# `  ( 0 ... A ) ) )
5447, 51, 533eqtrd 2466 . . . . 5  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i )  =  (
# `  ( 0 ... A ) ) )
5540, 54breqtrrd 4393 . . . 4  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =  1 )  ->  1  <_  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P )
) ^ i ) )
563, 4, 24, 30, 55letrd 9743 . . 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 6256 . . . . . . 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 4376 . . . . . 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 6256 . . . . . . 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 4376 . . . . . 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 9593 . . . . . . . . . 10  |-  1  e.  RR
6219adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  RR )
63 resubcl 9889 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( X `  ( L `
 P ) )  e.  RR )  -> 
( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
6461, 62, 63sylancr 667 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  RR )
6564adantr 466 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  RR )
6665leidd 10131 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  <_  ( 1  -  ( X `  ( L `  P )
) ) )
6764recnd 9620 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( X `  ( L `  P ) ) )  e.  CC )
6867adantr 466 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( X `
 ( L `  P ) ) )  e.  CC )
6968mulid2d 9612 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  =  ( 1  -  ( X `  ( L `  P ) ) ) )
70 nn0p1nn 10860 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e.  NN )
7131, 70syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN )
7271ad3antrrr 734 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( A  +  1 )  e.  NN )
73720expd 12382 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( 0 ^ ( A  +  1 ) )  =  0 )
74 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =  0 )  -> 
( X `  ( L `  P )
)  =  0 )
7574oveq1d 6264 . . . . . . . . . 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 2472 . . . . . . . . 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 10664 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
7831ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  NN0 )
79 expp1 12229 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  A  e.  NN0 )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u 1 ^ A )  x.  -u 1
) )
8077, 78, 79sylancr 667 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  ( ( -u
1 ^ A )  x.  -u 1 ) )
81 prmnn 14568 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P  e.  Prime  ->  P  e.  NN )
8215, 81syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  P  e.  NN )
8382, 31nnexpcld 12387 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( P ^ A
)  e.  NN )
8483nncnd 10576 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( P ^ A
)  e.  CC )
8584ad2antrr 730 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P ^ A )  e.  CC )
8685sqsqrtd 13444 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( sqr `  ( P ^ A ) ) ^ 2 )  =  ( P ^ A
) )
8786oveq2d 6265 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( ( sqr `  ( P ^ A
) ) ^ 2 ) )  =  ( P  pCnt  ( P ^ A ) ) )
8815ad2antrr 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  P  e.  Prime )
89 nnq 11228 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
9089adantl 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  QQ )
91 nnne0 10593 . . . . . . . . . . . . . . . . . . 19  |-  ( ( sqr `  ( P ^ A ) )  e.  NN  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
9291adantl 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  =/=  0 )
93 2z 10920 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  ZZ
9493a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  ZZ )
95 pcexp 14752 . . . . . . . . . . . . . . . . . 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 1269 . . . . . . . . . . . . . . . . 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 10989 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  e.  ZZ )
98 pcid 14765 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
9988, 97, 98syl2anc 665 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
10087, 96, 993eqtr3rd 2471 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  A  =  ( 2  x.  ( P  pCnt  ( sqr `  ( P ^ A ) ) ) ) )
101100oveq2d 6265 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( sqr `  ( P ^ A ) )  e.  NN )
10488, 103pccld 14743 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  NN0 )
105 2nn0 10837 . . . . . . . . . . . . . . . . 17  |-  2  e.  NN0
106105a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  2  e.  NN0 )
107102, 104, 106expmuld 12369 . . . . . . . . . . . . . . 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 12320 . . . . . . . . . . . . . . . . 17  |-  ( -u
1 ^ 2 )  =  1
109108oveq1i 6259 . . . . . . . . . . . . . . . 16  |-  ( (
-u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  ( 1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )
110104nn0zd 10989 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( P  pCnt  ( sqr `  ( P ^ A ) ) )  e.  ZZ )
111 1exp 12251 . . . . . . . . . . . . . . . . 17  |-  ( ( P  pCnt  ( sqr `  ( P ^ A
) ) )  e.  ZZ  ->  ( 1 ^ ( P  pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
112110, 111syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1 ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
113109, 112syl5eq 2474 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ 2 ) ^ ( P 
pCnt  ( sqr `  ( P ^ A ) ) ) )  =  1 )
114101, 107, 1133eqtrd 2466 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ A )  =  1 )
115114oveq1d 6264 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  ( 1  x.  -u 1 ) )
11677mulid2i 9597 . . . . . . . . . . . . 13  |-  ( 1  x.  -u 1 )  = 
-u 1
117115, 116syl6eq 2478 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( -u 1 ^ A
)  x.  -u 1
)  =  -u 1
)
11880, 117eqtrd 2462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  ( -u 1 ^ ( A  +  1 ) )  =  -u 1 )
119118adantr 466 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( -u 1 ^ ( A  +  1 ) )  =  -u 1
)
12019recnd 9620 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X `  ( L `  P )
)  e.  CC )
121120adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  e.  CC )
122121ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  CC )
123122negnegd 9928 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  =/=  1
)
125124ad2antrr 730 . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  X  e.  D )
130 eqid 2428 . . . . . . . . . . . . . . . . . . 19  |-  (Unit `  Z )  =  (Unit `  Z )
131126, 9, 127, 10, 130, 128, 18dchrn0 24120 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( X `  ( L `  P ) )  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
132131ad2antrr 730 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
)  =/=  0  <->  ( L `  P )  e.  (Unit `  Z )
) )
133132biimpa 486 . . . . . . . . . . . . . . . . . . 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 24130 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( abs `  ( X `  ( L `  P ) ) )  =  1 )
135 eqeq1 2432 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  ( X `
 ( L `  P ) ) )  =  ( X `  ( L `  P ) )  ->  ( ( abs `  ( X `  ( L `  P ) ) )  =  1  <-> 
( X `  ( L `  P )
)  =  1 ) )
136134, 135syl5ibcom 223 . . . . . . . . . . . . . . . . 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 2614 . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  e.  RR )
140139absord 13421 . . . . . . . . . . . . . . . 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 378 . . . . . . . . . . . . . . 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 2464 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u ( X `  ( L `  P )
)  =  1 )
144143negeqd 9820 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  ->  -u -u ( X `  ( L `  P )
)  =  -u 1
)
145123, 144eqtr3d 2464 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( X `  ( L `
 P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  /\  ( X `  ( L `
 P ) )  =/=  0 )  -> 
( X `  ( L `  P )
)  =  -u 1
)
146145oveq1d 6264 . . . . . . . . . 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 2472 . . . . . . . . 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 2688 . . . . . . . 8  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
( X `  ( L `  P )
) ^ ( A  +  1 ) )  =  ( X `  ( L `  P ) ) )
149148oveq2d 6265 . . . . . . 7  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) )  =  ( 1  -  ( X `  ( L `  P )
) ) )
15066, 69, 1493brtr4d 4397 . . . . . 6  |-  ( ( ( ph  /\  ( X `  ( L `  P ) )  =/=  1 )  /\  ( sqr `  ( P ^ A ) )  e.  NN )  ->  (
1  x.  ( 1  -  ( X `  ( L `  P ) ) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
15167mul02d 9782 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  =  0 )
152 peano2nn0 10861 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
15331, 152syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
15419, 153reexpcld 12383 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )
155154adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  RR )
156155recnd 9620 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  e.  CC )
157156abscld 13441 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  e.  RR )
158 1red 9609 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  e.  RR )
159155leabsd 13420 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  ( abs `  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) ) )
160153adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e. 
NN0 )
161121, 160absexpd 13457 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  =  ( ( abs `  ( X `
 ( L `  P ) ) ) ^ ( A  + 
1 ) ) )
162121abscld 13441 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  e.  RR )
163121absge0d 13449 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( abs `  ( X `
 ( L `  P ) ) ) )
164126, 127, 9, 10, 128, 18dchrabs2 24132 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( X `  ( L `  P ) ) )  <_  1 )
165164adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( X `  ( L `  P )
) )  <_  1
)
166 exple1 12282 . . . . . . . . . . . 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 1267 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( abs `  ( X `  ( L `  P ) ) ) ^ ( A  +  1 ) )  <_  1 )
168161, 167eqbrtrd 4387 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( abs `  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <_  1 )
169155, 157, 158, 159, 168letrd 9743 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
( A  +  1 ) )  <_  1
)
170 subge0 10078 . . . . . . . . . 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 667 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  <_  ( 1  -  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) ) )  <->  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) )  <_  1
) )
172169, 171mpbird 235 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <_  ( 1  -  ( ( X `  ( L `
 P ) ) ^ ( A  + 
1 ) ) ) )
173151, 172eqbrtrd 4387 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0  x.  ( 1  -  ( X `  ( L `  P )
) ) )  <_ 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
174173adantr 466 . . . . . 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 3889 . . . . 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 9594 . . . . . . . 8  |-  0  e.  RR
17761, 176keepel 3921 . . . . . . 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 9889 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( ( X `  ( L `  P ) ) ^ ( A  +  1 ) )  e.  RR )  -> 
( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) )  e.  RR )
18061, 155, 179sylancr 667 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 1  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  e.  RR )
181124necomd 2656 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  1  =/=  ( X `  ( L `
 P ) ) )
18262leabsd 13420 . . . . . . . . . 10  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  ( abs `  ( X `  ( L `  P ) ) ) )
18362, 162, 158, 182, 165letrd 9743 . . . . . . . . 9  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <_  1
)
18462, 158, 183leltned 9739 . . . . . . . 8  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  1  =/=  ( X `  ( L `  P ) ) ) )
185181, 184mpbird 235 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( X `  ( L `  P
) )  <  1
)
186 posdif 10058 . . . . . . . 8  |-  ( ( ( X `  ( L `  P )
)  e.  RR  /\  1  e.  RR )  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  ( 1  -  ( X `  ( L `  P )
) ) ) )
18762, 61, 186sylancl 666 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) )  <  1  <->  0  <  (
1  -  ( X `
 ( L `  P ) ) ) ) )
188185, 187mpbid 213 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  <  ( 1  -  ( X `
 ( L `  P ) ) ) )
189 lemuldiv 10437 . . . . . 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 1268 . . . . 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 213 . . . 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 10989 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
193 fzval3 11933 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
194192, 193syl 17 . . . . . . 7  |-  ( ph  ->  ( 0 ... A
)  =  ( 0..^ ( A  +  1 ) ) )
195194adantr 466 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( 0 ... A )  =  ( 0..^ ( A  +  1 ) ) )
196195sumeq1d 13710 . . . . 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 10835 . . . . . . 7  |-  0  e.  NN0
198197a1i 11 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  0  e.  NN0 )
199153, 32syl6eleq 2516 . . . . . . 7  |-  ( ph  ->  ( A  +  1 )  e.  ( ZZ>= ` 
0 ) )
200199adantr 466 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( A  +  1 )  e.  ( ZZ>= `  0 )
)
201121, 124, 198, 200geoserg 13867 . . . . 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 12360 . . . . . . 7  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( ( X `  ( L `  P ) ) ^
0 )  =  1 )
203202oveq1d 6264 . . . . . 6  |-  ( (
ph  /\  ( X `  ( L `  P
) )  =/=  1
)  ->  ( (
( X `  ( L `  P )
) ^ 0 )  -  ( ( X `
 ( L `  P ) ) ^
( A  +  1 ) ) )  =  ( 1  -  (
( X `  ( L `  P )
) ^ ( A  +  1 ) ) ) )
204203oveq1d 6264 . . . . 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 2466 . . . 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 4393 . . 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 2688 . 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 24285 . . . 4  |-  ( ( P ^ A )  e.  NN  ->  ( F `  ( P ^ A ) )  = 
sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
21183, 210syl 17 . . 3  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ k  e.  {
q  e.  NN  | 
q  ||  ( P ^ A ) }  ( X `  ( L `  k ) ) )
212 fveq2 5825 . . . . 5  |-  ( k  =  ( P ^
i )  ->  ( L `  k )  =  ( L `  ( P ^ i ) ) )
213212fveq2d 5829 . . . 4  |-  ( k  =  ( P ^
i )  ->  ( X `  ( L `  k ) )  =  ( X `  ( L `  ( P ^ i ) ) ) )
214 eqid 2428 . . . . . 6  |-  ( b  e.  ( 0 ... A )  |->  ( P ^ b ) )  =  ( b  e.  ( 0 ... A
)  |->  ( P ^
b ) )
215214dvdsppwf1o 24057 . . . . 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 665 . . . 4  |-  ( ph  ->  ( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) : ( 0 ... A ) -1-1-onto-> { q  e.  NN  | 
q  ||  ( P ^ A ) } )
217 oveq2 6257 . . . . . 6  |-  ( b  =  i  ->  ( P ^ b )  =  ( P ^ i
) )
218 ovex 6277 . . . . . 6  |-  ( P ^ b )  e. 
_V
219217, 214, 218fvmpt3i 5913 . . . . 5  |-  ( i  e.  ( 0 ... A )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
220219adantl 467 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
( b  e.  ( 0 ... A ) 
|->  ( P ^ b
) ) `  i
)  =  ( P ^ i ) )
2216adantr 466 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  X : (
Base `  Z ) --> RR )
222 elrabi 3168 . . . . . . . 8  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  NN )
223222nnzd 10990 . . . . . . 7  |-  ( k  e.  { q  e.  NN  |  q  ||  ( P ^ A ) }  ->  k  e.  ZZ )
224 ffvelrn 5979 . . . . . . 7  |-  ( ( L : ZZ --> ( Base `  Z )  /\  k  e.  ZZ )  ->  ( L `  k )  e.  ( Base `  Z
) )
22514, 223, 224syl2an 479 . . . . . 6  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( L `  k )  e.  (
Base `  Z )
)
226221, 225ffvelrnd 5982 . . . . 5  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  RR )
227226recnd 9620 . . . 4  |-  ( (
ph  /\  k  e.  { q  e.  NN  | 
q  ||  ( P ^ A ) } )  ->  ( X `  ( L `  k ) )  e.  CC )
228213, 5, 216, 220, 227fsumf1o 13732 . . 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 18964 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
230 eqid 2428 . . . . . . . . . . . 12  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
231230subrgsubm 17964 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
232229, 231mp1i 13 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ZZ  e.  (SubMnd `  (mulGrp ` fld ) ) )
23320adantl 467 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  i  e.  NN0 )
23417adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  P  e.  ZZ )
235 eqid 2428 . . . . . . . . . . 11  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
236 zringmpg 19005 . . . . . . . . . . . 12  |-  ( (mulGrp ` fld )s  ZZ )  =  (mulGrp ` ring )
237236eqcomi 2437 . . . . . . . . . . 11  |-  (mulGrp ` ring )  =  ( (mulGrp ` fld )s  ZZ )
238 eqid 2428 . . . . . . . . . . 11  |-  (.g `  (mulGrp ` ring ) )  =  (.g `  (mulGrp ` ring ) )
239235, 237, 238submmulg 16736 . . . . . . . . . 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 1264 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( i (.g `  (mulGrp ` ring ) ) P ) )
24182nncnd 10576 . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
242 cnfldexp 18944 . . . . . . . . . 10  |-  ( ( P  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^ i
) )
243241, 20, 242syl2an 479 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) P )  =  ( P ^
i ) )
244240, 243eqtr3d 2464 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` ring ) ) P )  =  ( P ^
i ) )
245244fveq2d 5829 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp ` ring ) ) P ) )  =  ( L `
 ( P ^
i ) ) )
2469zncrng 19057 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
247 crngring 17734 . . . . . . . . . . 11  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
2488, 246, 2473syl 18 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  Ring )
24911zrhrhm 19025 . . . . . . . . . 10  |-  ( Z  e.  Ring  ->  L  e.  (ring RingHom  Z ) )
250 eqid 2428 . . . . . . . . . . 11  |-  (mulGrp ` ring )  =  (mulGrp ` ring )
251 eqid 2428 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
252250, 251rhmmhm 17893 . . . . . . . . . 10  |-  ( L  e.  (ring RingHom  Z )  ->  L  e.  ( (mulGrp ` ring ) MndHom  (mulGrp `  Z
) ) )
253248, 249, 2523syl 18 . . . . . . . . 9  |-  ( ph  ->  L  e.  ( (mulGrp ` ring ) MndHom  (mulGrp `  Z )
) )
254253adantr 466 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  L  e.  ( (mulGrp ` ring ) MndHom  (mulGrp `  Z
) ) )
255 zringbas 18987 . . . . . . . . . 10  |-  ZZ  =  ( Base ` ring )
256250, 255mgpbas 17672 . . . . . . . . 9  |-  ZZ  =  ( Base `  (mulGrp ` ring ) )
257 eqid 2428 . . . . . . . . 9  |-  (.g `  (mulGrp `  Z ) )  =  (.g `  (mulGrp `  Z
) )
258256, 238, 257mhmmulg 16733 . . . . . . . 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 1264 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( i
(.g `  (mulGrp ` ring ) ) P ) )  =  ( i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )
260245, 259eqtr3d 2464 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  ( P ^ i ) )  =  ( i (.g `  (mulGrp `  Z )
) ( L `  P ) ) )
261260fveq2d 5829 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( X `  (
i (.g `  (mulGrp `  Z
) ) ( L `
 P ) ) ) )
262126, 9, 127dchrmhm 24111 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
263262, 128sseldi 3405 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
264263adantr 466 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
26518adantr 466 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( L `  P )  e.  ( Base `  Z
) )
266251, 10mgpbas 17672 . . . . . . 7  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
267266, 257, 235mhmmulg 16733 . . . . . 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 1264 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( i
(.g `  (mulGrp `  Z
) ) ( L `
 P ) ) )  =  ( i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) ) )
269 cnfldexp 18944 . . . . . 6  |-  ( ( ( X `  ( L `  P )
)  e.  CC  /\  i  e.  NN0 )  -> 
( i (.g `  (mulGrp ` fld ) ) ( X `  ( L `  P ) ) )  =  ( ( X `  ( L `  P )
) ^ i ) )
270120, 20, 269syl2an 479 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  (
i (.g `  (mulGrp ` fld ) ) ( X `
 ( L `  P ) ) )  =  ( ( X `
 ( L `  P ) ) ^
i ) )
271261, 268, 2703eqtrd 2466 . . . 4  |-  ( (
ph  /\  i  e.  ( 0 ... A
) )  ->  ( X `  ( L `  ( P ^ i
) ) )  =  ( ( X `  ( L `  P ) ) ^ i ) )
272271sumeq2dv 13712 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 0 ... A ) ( X `  ( L `  ( P ^ i ) ) )  =  sum_ i  e.  ( 0 ... A
) ( ( X `
 ( L `  P ) ) ^
i ) )
273211, 228, 2723eqtrd 2466 . 2  |-  ( ph  ->  ( F `  ( P ^ A ) )  =  sum_ i  e.  ( 0 ... A ) ( ( X `  ( L `  P ) ) ^ i ) )
274207, 273breqtrrd 4393 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   {crab 2718   (/)c0 3704   ifcif 3854   class class class wbr 4366    |-> cmpt 4425   -->wf 5540   -onto->wfo 5542   -1-1-onto->wf1o 5543   ` cfv 5544  (class class class)co 6249   Fincfn 7524   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    < clt 9626    <_ cle 9627    - cmin 9811   -ucneg 9812    / cdiv 10220   NNcn 10560   2c2 10610   NN0cn0 10820   ZZcz 10888   ZZ>=cuz 11110   QQcq 11215   ...cfz 11735  ..^cfzo 11866   ^cexp 12222   #chash 12465   sqrcsqrt 13240   abscabs 13241   sum_csu 13695    || cdvds 14248   Primecprime 14565    pCnt cpc 14729   Basecbs 15064   ↾s cress 15065   0gc0g 15281   MndHom cmhm 16523  SubMndcsubmnd 16524  .gcmg 16615  mulGrpcmgp 17666   Ringcrg 17723   CRingccrg 17724  Unitcui 17810   RingHom crh 17883  SubRingcsubrg 17947  ℂfldccnfld 18913  ℤringzring 18981   ZRHomczrh 19013  ℤ/nczn 19016  DChrcdchr 24102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-disj 4338  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-tpos 6928  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-omul 7142  df-er 7318  df-ec 7320  df-qs 7324  df-map 7429  df-pm 7430  df-ixp 7478  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fsupp 7837  df-fi 7878  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  df-acn 8328  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-ioo 11590  df-ioc 11591  df-ico 11592  df-icc 11593  df-fz 11736  df-fzo 11867  df-fl 11978  df-mod 12047  df-seq 12164  df-exp 12223  df-fac 12410  df-bc 12438  df-hash 12466  df-shft 13074  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-limsup 13469  df-clim 13495  df-rlim 13496  df-sum 13696  df-ef 14064  df-sin 14066  df-cos 14067  df-pi 14069  df-dvds 14249  df-gcd 14412  df-prm 14566  df-pc 14730  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-starv 15148  df-sca 15149  df-vsca 15150  df-ip 15151  df-tset 15152  df-ple 15153  df-ds 15155  df-unif 15156  df-hom 15157  df-cco 15158  df-rest 15264  df-topn 15265  df-0g 15283  df-gsum 15284  df-topgen 15285  df-pt 15286  df-prds 15289  df-xrs 15343  df-qtop 15349  df-imas 15350  df-qus 15352  df-xps 15353  df-mre 15435  df-mrc 15436  df-acs 15438  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-mhm 16525  df-submnd 16526  df-grp 16616  df-minusg 16617  df-sbg 16618  df-mulg 16619  df-subg 16757  df-nsg 16758  df-eqg 16759  df-ghm 16824  df-cntz 16914  df-od 17115  df-cmn 17375  df-abl 17376  df-mgp 17667  df-ur 17679  df-ring 17725  df-cring 17726  df-oppr 17794  df-dvdsr 17812  df-unit 17813  df-invr 17843  df-dvr 17854  df-rnghom 17886  df-drng 17920  df-subrg 17949  df-lmod 18036  df-lss 18099  df-lsp 18138  df-sra 18338  df-rgmod 18339  df-lidl 18340  df-rsp 18341  df-2idl 18399  df-psmet 18905  df-xmet 18906  df-met 18907  df-bl 18908  df-mopn 18909  df-fbas 18910  df-fg 18911  df-cnfld 18914  df-zring 18982  df-zrh 19017  df-zn 19020  df-top 19863  df-bases 19864  df-topon 19865  df-topsp 19866  df-cld 19976  df-ntr 19977  df-cls 19978  df-nei 20056  df-lp 20094  df-perf 20095  df-cn 20185  df-cnp 20186  df-haus 20273  df-tx 20519  df-hmeo 20712  df-fil 20803  df-fm 20895  df-flim 20896  df-flf 20897  df-xms 21277  df-ms 21278  df-tms 21279  df-cncf 21852  df-limc 22763  df-dv 22764  df-log 23448  df-cxp 23449  df-dchr 24103
This theorem is referenced by:  dchrisum0flblem2  24289  dchrisum0flb  24290
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