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Theorem vdwlem8 13311
Description: Lemma for vdw 13317. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdwlem8.r  |-  ( ph  ->  R  e.  Fin )
vdwlem8.k  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
vdwlem8.w  |-  ( ph  ->  W  e.  NN )
vdwlem8.f  |-  ( ph  ->  F : ( 1 ... ( 2  x.  W ) ) --> R )
vdwlem8.c  |-  C  e. 
_V
vdwlem8.a  |-  ( ph  ->  A  e.  NN )
vdwlem8.d  |-  ( ph  ->  D  e.  NN )
vdwlem8.s  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' G " { C } ) )
vdwlem8.g  |-  G  =  ( x  e.  ( 1 ... W ) 
|->  ( F `  (
x  +  W ) ) )
Assertion
Ref Expression
vdwlem8  |-  ( ph  -> 
<. 1 ,  K >. PolyAP 
F )
Distinct variable groups:    x, A    x, D    x, F    ph, x    x, C    x, K    x, W
Allowed substitution hints:    R( x)    G( x)

Proof of Theorem vdwlem8
Dummy variables  a 
d  i  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwlem8.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  NN )
21nncnd 9972 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
3 vdwlem8.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  NN )
43nncnd 9972 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
52, 4addcomd 9224 . . . . . . . 8  |-  ( ph  ->  ( A  +  D
)  =  ( D  +  A ) )
65oveq2d 6056 . . . . . . 7  |-  ( ph  ->  ( W  -  ( A  +  D )
)  =  ( W  -  ( D  +  A ) ) )
7 vdwlem8.w . . . . . . . . 9  |-  ( ph  ->  W  e.  NN )
87nncnd 9972 . . . . . . . 8  |-  ( ph  ->  W  e.  CC )
98, 4, 2subsub4d 9398 . . . . . . 7  |-  ( ph  ->  ( ( W  -  D )  -  A
)  =  ( W  -  ( D  +  A ) ) )
106, 9eqtr4d 2439 . . . . . 6  |-  ( ph  ->  ( W  -  ( A  +  D )
)  =  ( ( W  -  D )  -  A ) )
1110oveq2d 6056 . . . . 5  |-  ( ph  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  =  ( ( A  +  A )  +  ( ( W  -  D )  -  A ) ) )
128, 4subcld 9367 . . . . . 6  |-  ( ph  ->  ( W  -  D
)  e.  CC )
132, 2, 12ppncand 9407 . . . . 5  |-  ( ph  ->  ( ( A  +  A )  +  ( ( W  -  D
)  -  A ) )  =  ( A  +  ( W  -  D ) ) )
1411, 13eqtrd 2436 . . . 4  |-  ( ph  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  =  ( A  +  ( W  -  D ) ) )
151, 1nnaddcld 10002 . . . . 5  |-  ( ph  ->  ( A  +  A
)  e.  NN )
16 vdwlem8.s . . . . . . . 8  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' G " { C } ) )
17 cnvimass 5183 . . . . . . . . 9  |-  ( `' G " { C } )  C_  dom  G
18 fvex 5701 . . . . . . . . . 10  |-  ( F `
 ( x  +  W ) )  e. 
_V
19 vdwlem8.g . . . . . . . . . 10  |-  G  =  ( x  e.  ( 1 ... W ) 
|->  ( F `  (
x  +  W ) ) )
2018, 19dmmpti 5533 . . . . . . . . 9  |-  dom  G  =  ( 1 ... W )
2117, 20sseqtri 3340 . . . . . . . 8  |-  ( `' G " { C } )  C_  (
1 ... W )
2216, 21syl6ss 3320 . . . . . . 7  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( 1 ... W ) )
23 ssun2 3471 . . . . . . . . 9  |-  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D )  C_  ( { A }  u.  (
( A  +  D
) (AP `  ( K  -  1 ) ) D ) )
24 vdwlem8.k . . . . . . . . . . 11  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
25 uz2m1nn 10506 . . . . . . . . . . 11  |-  ( K  e.  ( ZZ>= `  2
)  ->  ( K  -  1 )  e.  NN )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( K  -  1 )  e.  NN )
271, 3nnaddcld 10002 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  D
)  e.  NN )
28 vdwapid1 13298 . . . . . . . . . 10  |-  ( ( ( K  -  1 )  e.  NN  /\  ( A  +  D
)  e.  NN  /\  D  e.  NN )  ->  ( A  +  D
)  e.  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D ) )
2926, 27, 3, 28syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( A  +  D
)  e.  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D ) )
3023, 29sseldi 3306 . . . . . . . 8  |-  ( ph  ->  ( A  +  D
)  e.  ( { A }  u.  (
( A  +  D
) (AP `  ( K  -  1 ) ) D ) ) )
31 eluz2b2 10504 . . . . . . . . . . . . . . 15  |-  ( K  e.  ( ZZ>= `  2
)  <->  ( K  e.  NN  /\  1  < 
K ) )
3231simplbi 447 . . . . . . . . . . . . . 14  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  NN )
3324, 32syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  NN )
3433nncnd 9972 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  CC )
35 ax-1cn 9004 . . . . . . . . . . . 12  |-  1  e.  CC
36 npcan 9270 . . . . . . . . . . . 12  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
3734, 35, 36sylancl 644 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
3837fveq2d 5691 . . . . . . . . . 10  |-  ( ph  ->  (AP `  ( ( K  -  1 )  +  1 ) )  =  (AP `  K
) )
3938oveqd 6057 . . . . . . . . 9  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( A (AP
`  K ) D ) )
4026nnnn0d 10230 . . . . . . . . . 10  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
41 vdwapun 13297 . . . . . . . . . 10  |-  ( ( ( K  -  1 )  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( A (AP `  ( ( K  -  1 )  +  1 ) ) D )  =  ( { A }  u.  ( ( A  +  D ) (AP `  ( K  -  1
) ) D ) ) )
4240, 1, 3, 41syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4339, 42eqtr3d 2438 . . . . . . . 8  |-  ( ph  ->  ( A (AP `  K ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4430, 43eleqtrrd 2481 . . . . . . 7  |-  ( ph  ->  ( A  +  D
)  e.  ( A (AP `  K ) D ) )
4522, 44sseldd 3309 . . . . . 6  |-  ( ph  ->  ( A  +  D
)  e.  ( 1 ... W ) )
46 elfzuz3 11012 . . . . . 6  |-  ( ( A  +  D )  e.  ( 1 ... W )  ->  W  e.  ( ZZ>= `  ( A  +  D ) ) )
47 uznn0sub 10473 . . . . . 6  |-  ( W  e.  ( ZZ>= `  ( A  +  D )
)  ->  ( W  -  ( A  +  D ) )  e. 
NN0 )
4845, 46, 473syl 19 . . . . 5  |-  ( ph  ->  ( W  -  ( A  +  D )
)  e.  NN0 )
49 nnnn0addcl 10207 . . . . 5  |-  ( ( ( A  +  A
)  e.  NN  /\  ( W  -  ( A  +  D )
)  e.  NN0 )  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  e.  NN )
5015, 48, 49syl2anc 643 . . . 4  |-  ( ph  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  e.  NN )
5114, 50eqeltrrd 2479 . . 3  |-  ( ph  ->  ( A  +  ( W  -  D ) )  e.  NN )
52 1nn 9967 . . . . . . . 8  |-  1  e.  NN
53 f1osng 5675 . . . . . . . 8  |-  ( ( 1  e.  NN  /\  D  e.  NN )  ->  { <. 1 ,  D >. } : { 1 } -1-1-onto-> { D } )
5452, 3, 53sylancr 645 . . . . . . 7  |-  ( ph  ->  { <. 1 ,  D >. } : { 1 } -1-1-onto-> { D } )
55 f1of 5633 . . . . . . 7  |-  ( {
<. 1 ,  D >. } : { 1 } -1-1-onto-> { D }  ->  {
<. 1 ,  D >. } : { 1 } --> { D }
)
5654, 55syl 16 . . . . . 6  |-  ( ph  ->  { <. 1 ,  D >. } : { 1 } --> { D }
)
573snssd 3903 . . . . . 6  |-  ( ph  ->  { D }  C_  NN )
58 fss 5558 . . . . . 6  |-  ( ( { <. 1 ,  D >. } : { 1 } --> { D }  /\  { D }  C_  NN )  ->  { <. 1 ,  D >. } : { 1 } --> NN )
5956, 57, 58syl2anc 643 . . . . 5  |-  ( ph  ->  { <. 1 ,  D >. } : { 1 } --> NN )
60 1z 10267 . . . . . . 7  |-  1  e.  ZZ
61 fzsn 11050 . . . . . . 7  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
6260, 61ax-mp 8 . . . . . 6  |-  ( 1 ... 1 )  =  { 1 }
6362feq2i 5545 . . . . 5  |-  ( {
<. 1 ,  D >. } : ( 1 ... 1 ) --> NN  <->  {
<. 1 ,  D >. } : { 1 } --> NN )
6459, 63sylibr 204 . . . 4  |-  ( ph  ->  { <. 1 ,  D >. } : ( 1 ... 1 ) --> NN )
65 nnex 9962 . . . . 5  |-  NN  e.  _V
66 ovex 6065 . . . . 5  |-  ( 1 ... 1 )  e. 
_V
6765, 66elmap 7001 . . . 4  |-  ( {
<. 1 ,  D >. }  e.  ( NN 
^m  ( 1 ... 1 ) )  <->  { <. 1 ,  D >. } : ( 1 ... 1 ) --> NN )
6864, 67sylibr 204 . . 3  |-  ( ph  ->  { <. 1 ,  D >. }  e.  ( NN 
^m  ( 1 ... 1 ) ) )
691, 7nnaddcld 10002 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  +  W
)  e.  NN )
7069adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  W )  e.  NN )
71 elfznn0 11039 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  m  e.  NN0 )
723nnnn0d 10230 . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  NN0 )
73 nn0mulcl 10212 . . . . . . . . . . . . . 14  |-  ( ( m  e.  NN0  /\  D  e.  NN0 )  -> 
( m  x.  D
)  e.  NN0 )
7471, 72, 73syl2anr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
m  x.  D )  e.  NN0 )
75 nnnn0addcl 10207 . . . . . . . . . . . . 13  |-  ( ( ( A  +  W
)  e.  NN  /\  ( m  x.  D
)  e.  NN0 )  ->  ( ( A  +  W )  +  ( m  x.  D ) )  e.  NN )
7670, 74, 75syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  e.  NN )
77 nnuz 10477 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
7876, 77syl6eleq 2494 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  e.  ( ZZ>= `  1
) )
7916adantr 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A (AP `  K ) D )  C_  ( `' G " { C } ) )
80 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  ( A  +  ( m  x.  D ) )  =  ( A  +  ( m  x.  D ) )
81 oveq1 6047 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  m  ->  (
n  x.  D )  =  ( m  x.  D ) )
8281oveq2d 6056 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  m  ->  ( A  +  ( n  x.  D ) )  =  ( A  +  ( m  x.  D ) ) )
8382eqeq2d 2415 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  m  ->  (
( A  +  ( m  x.  D ) )  =  ( A  +  ( n  x.  D ) )  <->  ( A  +  ( m  x.  D ) )  =  ( A  +  ( m  x.  D ) ) ) )
8483rspcev 3012 . . . . . . . . . . . . . . . . . 18  |-  ( ( m  e.  ( 0 ... ( K  - 
1 ) )  /\  ( A  +  (
m  x.  D ) )  =  ( A  +  ( m  x.  D ) ) )  ->  E. n  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( m  x.  D ) )  =  ( A  +  ( n  x.  D ) ) )
8580, 84mpan2 653 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  E. n  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( m  x.  D
) )  =  ( A  +  ( n  x.  D ) ) )
8633nnnn0d 10230 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  K  e.  NN0 )
87 vdwapval 13296 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  (
( A  +  ( m  x.  D ) )  e.  ( A (AP `  K ) D )  <->  E. n  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( m  x.  D
) )  =  ( A  +  ( n  x.  D ) ) ) )
8886, 1, 3, 87syl3anc 1184 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( A  +  ( m  x.  D
) )  e.  ( A (AP `  K
) D )  <->  E. n  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( m  x.  D
) )  =  ( A  +  ( n  x.  D ) ) ) )
8988biimpar 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  E. n  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( m  x.  D
) )  =  ( A  +  ( n  x.  D ) ) )  ->  ( A  +  ( m  x.  D ) )  e.  ( A (AP `  K ) D ) )
9085, 89sylan2 461 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  ( m  x.  D ) )  e.  ( A (AP `  K ) D ) )
9179, 90sseldd 3309 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  ( m  x.  D ) )  e.  ( `' G " { C } ) )
9218, 19fnmpti 5532 . . . . . . . . . . . . . . . 16  |-  G  Fn  ( 1 ... W
)
93 fniniseg 5810 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  ( 1 ... W )  ->  (
( A  +  ( m  x.  D ) )  e.  ( `' G " { C } )  <->  ( ( A  +  ( m  x.  D ) )  e.  ( 1 ... W
)  /\  ( G `  ( A  +  ( m  x.  D ) ) )  =  C ) ) )
9492, 93ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( ( A  +  ( m  x.  D ) )  e.  ( `' G " { C } )  <-> 
( ( A  +  ( m  x.  D
) )  e.  ( 1 ... W )  /\  ( G `  ( A  +  (
m  x.  D ) ) )  =  C ) )
9591, 94sylib 189 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  ( m  x.  D ) )  e.  ( 1 ... W )  /\  ( G `  ( A  +  ( m  x.  D ) ) )  =  C ) )
9695simpld 446 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  ( m  x.  D ) )  e.  ( 1 ... W
) )
97 elfzuz3 11012 . . . . . . . . . . . . 13  |-  ( ( A  +  ( m  x.  D ) )  e.  ( 1 ... W )  ->  W  e.  ( ZZ>= `  ( A  +  ( m  x.  D ) ) ) )
98 eluzelz 10452 . . . . . . . . . . . . . 14  |-  ( W  e.  ( ZZ>= `  ( A  +  ( m  x.  D ) ) )  ->  W  e.  ZZ )
99 eluzadd 10470 . . . . . . . . . . . . . 14  |-  ( ( W  e.  ( ZZ>= `  ( A  +  (
m  x.  D ) ) )  /\  W  e.  ZZ )  ->  ( W  +  W )  e.  ( ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
10098, 99mpdan 650 . . . . . . . . . . . . 13  |-  ( W  e.  ( ZZ>= `  ( A  +  ( m  x.  D ) ) )  ->  ( W  +  W )  e.  (
ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
10196, 97, 1003syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( W  +  W )  e.  ( ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
10282timesd 10166 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  W
)  =  ( W  +  W ) )
103102adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
2  x.  W )  =  ( W  +  W ) )
1042adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  A  e.  CC )
1058adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  W  e.  CC )
10674nn0cnd 10232 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
m  x.  D )  e.  CC )
107104, 105, 106add32d 9244 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  =  ( ( A  +  ( m  x.  D ) )  +  W ) )
108107fveq2d 5691 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( ZZ>=
`  ( ( A  +  W )  +  ( m  x.  D
) ) )  =  ( ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
109101, 103, 1083eltr4d 2485 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
2  x.  W )  e.  ( ZZ>= `  (
( A  +  W
)  +  ( m  x.  D ) ) ) )
110 elfzuzb 11009 . . . . . . . . . . 11  |-  ( ( ( A  +  W
)  +  ( m  x.  D ) )  e.  ( 1 ... ( 2  x.  W
) )  <->  ( (
( A  +  W
)  +  ( m  x.  D ) )  e.  ( ZZ>= `  1
)  /\  ( 2  x.  W )  e.  ( ZZ>= `  ( ( A  +  W )  +  ( m  x.  D ) ) ) ) )
11178, 109, 110sylanbrc 646 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  e.  ( 1 ... ( 2  x.  W
) ) )
112107fveq2d 5691 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) )  =  ( F `  ( ( A  +  ( m  x.  D
) )  +  W
) ) )
113 oveq1 6047 . . . . . . . . . . . . . 14  |-  ( x  =  ( A  +  ( m  x.  D
) )  ->  (
x  +  W )  =  ( ( A  +  ( m  x.  D ) )  +  W ) )
114113fveq2d 5691 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +  ( m  x.  D
) )  ->  ( F `  ( x  +  W ) )  =  ( F `  (
( A  +  ( m  x.  D ) )  +  W ) ) )
115 fvex 5701 . . . . . . . . . . . . 13  |-  ( F `
 ( ( A  +  ( m  x.  D ) )  +  W ) )  e. 
_V
116114, 19, 115fvmpt 5765 . . . . . . . . . . . 12  |-  ( ( A  +  ( m  x.  D ) )  e.  ( 1 ... W )  ->  ( G `  ( A  +  ( m  x.  D ) ) )  =  ( F `  ( ( A  +  ( m  x.  D
) )  +  W
) ) )
11796, 116syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( G `  ( A  +  ( m  x.  D ) ) )  =  ( F `  ( ( A  +  ( m  x.  D
) )  +  W
) ) )
11895simprd 450 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( G `  ( A  +  ( m  x.  D ) ) )  =  C )
119112, 117, 1183eqtr2d 2442 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) )  =  C )
120111, 119jca 519 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( ( A  +  W )  +  ( m  x.  D ) )  e.  ( 1 ... ( 2  x.  W ) )  /\  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) )  =  C ) )
121 eleq1 2464 . . . . . . . . . 10  |-  ( x  =  ( ( A  +  W )  +  ( m  x.  D
) )  ->  (
x  e.  ( 1 ... ( 2  x.  W ) )  <->  ( ( A  +  W )  +  ( m  x.  D ) )  e.  ( 1 ... (
2  x.  W ) ) ) )
122 fveq2 5687 . . . . . . . . . . 11  |-  ( x  =  ( ( A  +  W )  +  ( m  x.  D
) )  ->  ( F `  x )  =  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) ) )
123122eqeq1d 2412 . . . . . . . . . 10  |-  ( x  =  ( ( A  +  W )  +  ( m  x.  D
) )  ->  (
( F `  x
)  =  C  <->  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) )  =  C ) )
124121, 123anbi12d 692 . . . . . . . . 9  |-  ( x  =  ( ( A  +  W )  +  ( m  x.  D
) )  ->  (
( x  e.  ( 1 ... ( 2  x.  W ) )  /\  ( F `  x )  =  C )  <->  ( ( ( A  +  W )  +  ( m  x.  D ) )  e.  ( 1 ... (
2  x.  W ) )  /\  ( F `
 ( ( A  +  W )  +  ( m  x.  D
) ) )  =  C ) ) )
125120, 124syl5ibrcom 214 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
x  =  ( ( A  +  W )  +  ( m  x.  D ) )  -> 
( x  e.  ( 1 ... ( 2  x.  W ) )  /\  ( F `  x )  =  C ) ) )
126125rexlimdva 2790 . . . . . . 7  |-  ( ph  ->  ( E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( ( A  +  W
)  +  ( m  x.  D ) )  ->  ( x  e.  ( 1 ... (
2  x.  W ) )  /\  ( F `
 x )  =  C ) ) )
127 vdwapval 13296 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  ( A  +  W
)  e.  NN  /\  D  e.  NN )  ->  ( x  e.  ( ( A  +  W
) (AP `  K
) D )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( ( A  +  W
)  +  ( m  x.  D ) ) ) )
12886, 69, 3, 127syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ( A  +  W
) (AP `  K
) D )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( ( A  +  W
)  +  ( m  x.  D ) ) ) )
129 vdwlem8.f . . . . . . . 8  |-  ( ph  ->  F : ( 1 ... ( 2  x.  W ) ) --> R )
130 ffn 5550 . . . . . . . 8  |-  ( F : ( 1 ... ( 2  x.  W
) ) --> R  ->  F  Fn  ( 1 ... ( 2  x.  W ) ) )
131 fniniseg 5810 . . . . . . . 8  |-  ( F  Fn  ( 1 ... ( 2  x.  W
) )  ->  (
x  e.  ( `' F " { C } )  <->  ( x  e.  ( 1 ... (
2  x.  W ) )  /\  ( F `
 x )  =  C ) ) )
132129, 130, 1313syl 19 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( `' F " { C } )  <->  ( x  e.  ( 1 ... (
2  x.  W ) )  /\  ( F `
 x )  =  C ) ) )
133126, 128, 1323imtr4d 260 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( A  +  W
) (AP `  K
) D )  ->  x  e.  ( `' F " { C }
) ) )
134133ssrdv 3314 . . . . 5  |-  ( ph  ->  ( ( A  +  W ) (AP `  K ) D ) 
C_  ( `' F " { C } ) )
135 fvsng 5886 . . . . . . . . 9  |-  ( ( 1  e.  NN  /\  D  e.  NN )  ->  ( { <. 1 ,  D >. } `  1
)  =  D )
13652, 3, 135sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( { <. 1 ,  D >. } `  1
)  =  D )
137136oveq2d 6056 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) )  =  ( ( A  +  ( W  -  D ) )  +  D ) )
1382, 12, 4addassd 9066 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( W  -  D
) )  +  D
)  =  ( A  +  ( ( W  -  D )  +  D ) ) )
1398, 4npcand 9371 . . . . . . . 8  |-  ( ph  ->  ( ( W  -  D )  +  D
)  =  W )
140139oveq2d 6056 . . . . . . 7  |-  ( ph  ->  ( A  +  ( ( W  -  D
)  +  D ) )  =  ( A  +  W ) )
141137, 138, 1403eqtrd 2440 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) )  =  ( A  +  W ) )
142141, 136oveq12d 6058 . . . . 5  |-  ( ph  ->  ( ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) (AP `  K ) ( {
<. 1 ,  D >. } `  1 ) )  =  ( ( A  +  W ) (AP `  K ) D ) )
143141fveq2d 5691 . . . . . . . 8  |-  ( ph  ->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) )  =  ( F `  ( A  +  W ) ) )
144 vdwapid1 13298 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  A  e.  NN  /\  D  e.  NN )  ->  A  e.  ( A (AP `  K ) D ) )
14533, 1, 3, 144syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ( A (AP `  K ) D ) )
14616, 145sseldd 3309 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( `' G " { C } ) )
147 fniniseg 5810 . . . . . . . . . . . 12  |-  ( G  Fn  ( 1 ... W )  ->  ( A  e.  ( `' G " { C }
)  <->  ( A  e.  ( 1 ... W
)  /\  ( G `  A )  =  C ) ) )
14892, 147ax-mp 8 . . . . . . . . . . 11  |-  ( A  e.  ( `' G " { C } )  <-> 
( A  e.  ( 1 ... W )  /\  ( G `  A )  =  C ) )
149146, 148sylib 189 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( 1 ... W )  /\  ( G `  A )  =  C ) )
150149simpld 446 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 1 ... W ) )
151 oveq1 6047 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x  +  W )  =  ( A  +  W ) )
152151fveq2d 5691 . . . . . . . . . 10  |-  ( x  =  A  ->  ( F `  ( x  +  W ) )  =  ( F `  ( A  +  W )
) )
153 fvex 5701 . . . . . . . . . 10  |-  ( F `
 ( A  +  W ) )  e. 
_V
154152, 19, 153fvmpt 5765 . . . . . . . . 9  |-  ( A  e.  ( 1 ... W )  ->  ( G `  A )  =  ( F `  ( A  +  W
) ) )
155150, 154syl 16 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  =  ( F `
 ( A  +  W ) ) )
156149simprd 450 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  =  C )
157143, 155, 1563eqtr2d 2442 . . . . . . 7  |-  ( ph  ->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) )  =  C )
158157sneqd 3787 . . . . . 6  |-  ( ph  ->  { ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) }  =  { C } )
159158imaeq2d 5162 . . . . 5  |-  ( ph  ->  ( `' F " { ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) } )  =  ( `' F " { C } ) )
160134, 142, 1593sstr4d 3351 . . . 4  |-  ( ph  ->  ( ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) (AP `  K ) ( {
<. 1 ,  D >. } `  1 ) )  C_  ( `' F " { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) ) } ) )
161160ralrimivw 2750 . . 3  |-  ( ph  ->  A. i  e.  ( 1 ... 1 ) ( ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) (AP `  K ) ( {
<. 1 ,  D >. } `  1 ) )  C_  ( `' F " { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) ) } ) )
162157mpteq2dv 4256 . . . . . . . 8  |-  ( ph  ->  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) )  =  ( i  e.  ( 1 ... 1 ) 
|->  C ) )
163 fconstmpt 4880 . . . . . . . 8  |-  ( ( 1 ... 1 )  X.  { C }
)  =  ( i  e.  ( 1 ... 1 )  |->  C )
164162, 163syl6eqr 2454 . . . . . . 7  |-  ( ph  ->  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) )  =  ( ( 1 ... 1 )  X.  { C } ) )
165164rneqd 5056 . . . . . 6  |-  ( ph  ->  ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) )  =  ran  ( ( 1 ... 1 )  X. 
{ C } ) )
166 elfz3 11023 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  ( 1 ... 1
) )
167 ne0i 3594 . . . . . . . 8  |-  ( 1  e.  ( 1 ... 1 )  ->  (
1 ... 1 )  =/=  (/) )
16860, 166, 167mp2b 10 . . . . . . 7  |-  ( 1 ... 1 )  =/=  (/)
169 rnxp 5258 . . . . . . 7  |-  ( ( 1 ... 1 )  =/=  (/)  ->  ran  ( ( 1 ... 1 )  X.  { C }
)  =  { C } )
170168, 169ax-mp 8 . . . . . 6  |-  ran  (
( 1 ... 1
)  X.  { C } )  =  { C }
171165, 170syl6eq 2452 . . . . 5  |-  ( ph  ->  ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) )  =  { C } )
172171fveq2d 5691 . . . 4  |-  ( ph  ->  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) ) )  =  ( # `  { C } ) )
173 vdwlem8.c . . . . 5  |-  C  e. 
_V
174 hashsng 11602 . . . . 5  |-  ( C  e.  _V  ->  ( # `
 { C }
)  =  1 )
175173, 174ax-mp 8 . . . 4  |-  ( # `  { C } )  =  1
176172, 175syl6eq 2452 . . 3  |-  ( ph  ->  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) ) )  =  1 )
177 oveq1 6047 . . . . . . . 8  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
a  +  ( d `
 i ) )  =  ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) )
178177oveq1d 6055 . . . . . . 7  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  =  ( ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) )
179177fveq2d 5691 . . . . . . . . 9  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  ( F `  ( a  +  ( d `  i ) ) )  =  ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) ) )
180179sneqd 3787 . . . . . . . 8  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  { ( F `  ( a  +  ( d `  i ) ) ) }  =  { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) } )
181180imaeq2d 5162 . . . . . . 7  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) } ) )
182178, 181sseq12d 3337 . . . . . 6  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  <-> 
( ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) ) } ) ) )
183182ralbidv 2686 . . . . 5  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  ( A. i  e.  (
1 ... 1 ) ( ( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  <->  A. i  e.  ( 1 ... 1
) ( ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' F " { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) ) } ) ) )
184179mpteq2dv 4256 . . . . . . . 8  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
i  e.  ( 1 ... 1 )  |->  ( F `  ( a  +  ( d `  i ) ) ) )  =  ( i  e.  ( 1 ... 1 )  |->  ( F `
 ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) ) ) )
185184rneqd 5056 . . . . . . 7  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
a  +  ( d `
 i ) ) ) )  =  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( d `
 i ) ) ) ) )
186185fveq2d 5691 . . . . . 6  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  ( # `
 ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )  =  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( d `
 i ) ) ) ) ) )
187186eqeq1d 2412 . . . . 5  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
( # `  ran  (
i  e.  ( 1 ... 1 )  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )  =  1  <-> 
( # `  ran  (
i  e.  ( 1 ... 1 )  |->  ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) ) )  =  1 ) )
188183, 187anbi12d 692 . . . 4  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
( A. i  e.  ( 1 ... 1
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' F " { ( F `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( a  +  ( d `  i ) ) ) ) )  =  1 )  <->  ( A. i  e.  ( 1 ... 1 ) ( ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  C_  ( `' F " { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) } )  /\  ( # `
 ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) ) ) )  =  1 ) ) )
189 fveq1 5686 . . . . . . . . . 10  |-  ( d  =  { <. 1 ,  D >. }  ->  (
d `  i )  =  ( { <. 1 ,  D >. } `
 i ) )
190 elfz1eq 11024 . . . . . . . . . . 11  |-  ( i  e.  ( 1 ... 1 )  ->  i  =  1 )
191190fveq2d 5691 . . . . . . . . . 10  |-  ( i  e.  ( 1 ... 1 )  ->  ( { <. 1 ,  D >. } `  i )  =  ( { <. 1 ,  D >. } `
 1 ) )
192189, 191sylan9eq 2456 . . . . . . . . 9  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( d `  i )  =  ( { <. 1 ,  D >. } `  1 ) )
193192oveq2d 6056 . . . . . . . 8  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( ( A  +  ( W  -  D ) )  +  ( d `  i
) )  =  ( ( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) )
194193, 192oveq12d 6058 . . . . . . 7  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( (
( A  +  ( W  -  D ) )  +  ( d `
 i ) ) (AP `  K ) ( d `  i
) )  =  ( ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) (AP `  K
) ( { <. 1 ,  D >. } `
 1 ) ) )
195193fveq2d 5691 . . . . . . . . 9  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) )  =  ( F `  ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `
 1 ) ) ) )
196195sneqd 3787 . . . . . . . 8  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) ) }  =  { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) ) } )
197196imaeq2d 5162 . . . . . . 7  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( `' F " { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) ) } )  =  ( `' F " { ( F `  ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `
 1 ) ) ) } ) )
198194, 197sseq12d 3337 . . . . . 6  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( (
( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  C_  ( `' F " { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) } )  <->  ( (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) (AP `  K
) ( { <. 1 ,  D >. } `
 1 ) ) 
C_  ( `' F " { ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) } ) ) )
199198ralbidva 2682 . . . . 5  |-  ( d  =  { <. 1 ,  D >. }  ->  ( A. i  e.  (
1 ... 1 ) ( ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  C_  ( `' F " { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) } )  <->  A. i  e.  ( 1 ... 1
) ( ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `
 1 ) ) (AP `  K ) ( { <. 1 ,  D >. } `  1
) )  C_  ( `' F " { ( F `  ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `
 1 ) ) ) } ) ) )
200195mpteq2dva 4255 . . . . . . . 8  |-  ( d  =  { <. 1 ,  D >. }  ->  (
i  e.  ( 1 ... 1 )  |->  ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) )  =  ( i  e.  ( 1 ... 1 )  |->  ( F `
 ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) ) ) )
201200rneqd 5056 . . . . . . 7  |-  ( d  =  { <. 1 ,  D >. }  ->  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( d `
 i ) ) ) )  =  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) ) )
202201fveq2d 5691 . . . . . 6  |-  ( d  =  { <. 1 ,  D >. }  ->  ( # `
 ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) ) ) )  =  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) ) ) )
203202eqeq1d 2412 . . . . 5  |-  ( d  =  { <. 1 ,  D >. }  ->  (
( # `  ran  (
i  e.  ( 1 ... 1 )  |->  ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) ) )  =  1  <-> 
( # `  ran  (
i  e.  ( 1 ... 1 )  |->  ( F `  ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `
 1 ) ) ) ) )  =  1 ) )
204199, 203anbi12d 692 . . . 4  |-  ( d  =  { <. 1 ,  D >. }  ->  (
( A. i  e.  ( 1 ... 1
) ( ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' F " { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) ) ) )  =  1 )  <->  ( A. i  e.  ( 1 ... 1 ) ( ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) (AP `  K
) ( { <. 1 ,  D >. } `
 1 ) ) 
C_  ( `' F " { ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) ) )  =  1 ) ) )
205188, 204rspc2ev 3020 . . 3  |-  ( ( ( A  +  ( W  -  D ) )  e.  NN  /\  {
<. 1 ,  D >. }  e.  ( NN 
^m  ( 1 ... 1 ) )  /\  ( A. i  e.  ( 1 ... 1 ) ( ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) (AP `  K ) ( {
<. 1 ,  D >. } `  1 ) )  C_  ( `' F " { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) ) )  =  1 ) )  ->  E. a  e.  NN  E. d  e.  ( NN 
^m  ( 1 ... 1 ) ) ( A. i  e.  ( 1 ... 1 ) ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
a  +  ( d `
 i ) ) ) ) )  =  1 ) )
20651, 68, 161, 176, 205syl112anc 1188 . 2  |-  ( ph  ->  E. a  e.  NN  E. d  e.  ( NN 
^m  ( 1 ... 1 ) ) ( A. i  e.  ( 1 ... 1 ) ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
a  +  ( d `
 i ) ) ) ) )  =  1 ) )
207 ovex 6065 . . 3  |-  ( 1 ... ( 2  x.  W ) )  e. 
_V
20852a1i 11 . . 3  |-  ( ph  ->  1  e.  NN )
209 eqid 2404 . . 3  |-  ( 1 ... 1 )  =  ( 1 ... 1
)
210207, 86, 129, 208, 209vdwpc 13303 . 2  |-  ( ph  ->  ( <. 1 ,  K >. PolyAP 
F  <->  E. a  e.  NN  E. d  e.  ( NN 
^m  ( 1 ... 1 ) ) ( A. i  e.  ( 1 ... 1 ) ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
a  +  ( d `
 i ) ) ) ) )  =  1 ) ) )
211206, 210mpbird 224 1  |-  ( ph  -> 
<. 1 ,  K >. PolyAP 
F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916    u. cun 3278    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   #chash 11573  APcvdwa 13288   PolyAP cvdwp 13290
This theorem is referenced by:  vdwlem10  13313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-vdwap 13291  df-vdwpc 13293
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