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Theorem vdwlem8 14364
Description: Lemma for vdw 14370. (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 10551 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
3 vdwlem8.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  NN )
43nncnd 10551 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
52, 4addcomd 9780 . . . . . . . 8  |-  ( ph  ->  ( A  +  D
)  =  ( D  +  A ) )
65oveq2d 6299 . . . . . . 7  |-  ( ph  ->  ( W  -  ( A  +  D )
)  =  ( W  -  ( D  +  A ) ) )
7 vdwlem8.w . . . . . . . . 9  |-  ( ph  ->  W  e.  NN )
87nncnd 10551 . . . . . . . 8  |-  ( ph  ->  W  e.  CC )
98, 4, 2subsub4d 9960 . . . . . . 7  |-  ( ph  ->  ( ( W  -  D )  -  A
)  =  ( W  -  ( D  +  A ) ) )
106, 9eqtr4d 2511 . . . . . 6  |-  ( ph  ->  ( W  -  ( A  +  D )
)  =  ( ( W  -  D )  -  A ) )
1110oveq2d 6299 . . . . 5  |-  ( ph  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  =  ( ( A  +  A )  +  ( ( W  -  D )  -  A ) ) )
128, 4subcld 9929 . . . . . 6  |-  ( ph  ->  ( W  -  D
)  e.  CC )
132, 2, 12ppncand 9969 . . . . 5  |-  ( ph  ->  ( ( A  +  A )  +  ( ( W  -  D
)  -  A ) )  =  ( A  +  ( W  -  D ) ) )
1411, 13eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  =  ( A  +  ( W  -  D ) ) )
151, 1nnaddcld 10581 . . . . 5  |-  ( ph  ->  ( A  +  A
)  e.  NN )
16 vdwlem8.s . . . . . . . 8  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' G " { C } ) )
17 cnvimass 5356 . . . . . . . . 9  |-  ( `' G " { C } )  C_  dom  G
18 fvex 5875 . . . . . . . . . 10  |-  ( F `
 ( x  +  W ) )  e. 
_V
19 vdwlem8.g . . . . . . . . . 10  |-  G  =  ( x  e.  ( 1 ... W ) 
|->  ( F `  (
x  +  W ) ) )
2018, 19dmmpti 5709 . . . . . . . . 9  |-  dom  G  =  ( 1 ... W )
2117, 20sseqtri 3536 . . . . . . . 8  |-  ( `' G " { C } )  C_  (
1 ... W )
2216, 21syl6ss 3516 . . . . . . 7  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( 1 ... W ) )
23 ssun2 3668 . . . . . . . . 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 11155 . . . . . . . . . . 11  |-  ( K  e.  ( ZZ>= `  2
)  ->  ( K  -  1 )  e.  NN )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( K  -  1 )  e.  NN )
271, 3nnaddcld 10581 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  D
)  e.  NN )
28 vdwapid1 14351 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ph  ->  ( A  +  D
)  e.  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D ) )
3023, 29sseldi 3502 . . . . . . . 8  |-  ( ph  ->  ( A  +  D
)  e.  ( { A }  u.  (
( A  +  D
) (AP `  ( K  -  1 ) ) D ) ) )
31 eluz2b2 11153 . . . . . . . . . . . . . . 15  |-  ( K  e.  ( ZZ>= `  2
)  <->  ( K  e.  NN  /\  1  < 
K ) )
3231simplbi 460 . . . . . . . . . . . . . 14  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  NN )
3324, 32syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  NN )
3433nncnd 10551 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  CC )
35 ax-1cn 9549 . . . . . . . . . . . 12  |-  1  e.  CC
36 npcan 9828 . . . . . . . . . . . 12  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
3734, 35, 36sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
3837fveq2d 5869 . . . . . . . . . 10  |-  ( ph  ->  (AP `  ( ( K  -  1 )  +  1 ) )  =  (AP `  K
) )
3938oveqd 6300 . . . . . . . . 9  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( A (AP
`  K ) D ) )
4026nnnn0d 10851 . . . . . . . . . 10  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
41 vdwapun 14350 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4339, 42eqtr3d 2510 . . . . . . . 8  |-  ( ph  ->  ( A (AP `  K ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4430, 43eleqtrrd 2558 . . . . . . 7  |-  ( ph  ->  ( A  +  D
)  e.  ( A (AP `  K ) D ) )
4522, 44sseldd 3505 . . . . . 6  |-  ( ph  ->  ( A  +  D
)  e.  ( 1 ... W ) )
46 elfzuz3 11684 . . . . . 6  |-  ( ( A  +  D )  e.  ( 1 ... W )  ->  W  e.  ( ZZ>= `  ( A  +  D ) ) )
47 uznn0sub 11112 . . . . . 6  |-  ( W  e.  ( ZZ>= `  ( A  +  D )
)  ->  ( W  -  ( A  +  D ) )  e. 
NN0 )
4845, 46, 473syl 20 . . . . 5  |-  ( ph  ->  ( W  -  ( A  +  D )
)  e.  NN0 )
49 nnnn0addcl 10825 . . . . 5  |-  ( ( ( A  +  A
)  e.  NN  /\  ( W  -  ( A  +  D )
)  e.  NN0 )  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  e.  NN )
5015, 48, 49syl2anc 661 . . . 4  |-  ( ph  ->  ( ( A  +  A )  +  ( W  -  ( A  +  D ) ) )  e.  NN )
5114, 50eqeltrrd 2556 . . 3  |-  ( ph  ->  ( A  +  ( W  -  D ) )  e.  NN )
52 1nn 10546 . . . . . . . 8  |-  1  e.  NN
53 f1osng 5853 . . . . . . . 8  |-  ( ( 1  e.  NN  /\  D  e.  NN )  ->  { <. 1 ,  D >. } : { 1 } -1-1-onto-> { D } )
5452, 3, 53sylancr 663 . . . . . . 7  |-  ( ph  ->  { <. 1 ,  D >. } : { 1 } -1-1-onto-> { D } )
55 f1of 5815 . . . . . . 7  |-  ( {
<. 1 ,  D >. } : { 1 } -1-1-onto-> { D }  ->  {
<. 1 ,  D >. } : { 1 } --> { D }
)
5654, 55syl 16 . . . . . 6  |-  ( ph  ->  { <. 1 ,  D >. } : { 1 } --> { D }
)
573snssd 4172 . . . . . 6  |-  ( ph  ->  { D }  C_  NN )
58 fss 5738 . . . . . 6  |-  ( ( { <. 1 ,  D >. } : { 1 } --> { D }  /\  { D }  C_  NN )  ->  { <. 1 ,  D >. } : { 1 } --> NN )
5956, 57, 58syl2anc 661 . . . . 5  |-  ( ph  ->  { <. 1 ,  D >. } : { 1 } --> NN )
60 1z 10893 . . . . . . 7  |-  1  e.  ZZ
61 fzsn 11724 . . . . . . 7  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
6260, 61ax-mp 5 . . . . . 6  |-  ( 1 ... 1 )  =  { 1 }
6362feq2i 5723 . . . . 5  |-  ( {
<. 1 ,  D >. } : ( 1 ... 1 ) --> NN  <->  {
<. 1 ,  D >. } : { 1 } --> NN )
6459, 63sylibr 212 . . . 4  |-  ( ph  ->  { <. 1 ,  D >. } : ( 1 ... 1 ) --> NN )
65 nnex 10541 . . . . 5  |-  NN  e.  _V
66 ovex 6308 . . . . 5  |-  ( 1 ... 1 )  e. 
_V
6765, 66elmap 7447 . . . 4  |-  ( {
<. 1 ,  D >. }  e.  ( NN 
^m  ( 1 ... 1 ) )  <->  { <. 1 ,  D >. } : ( 1 ... 1 ) --> NN )
6864, 67sylibr 212 . . 3  |-  ( ph  ->  { <. 1 ,  D >. }  e.  ( NN 
^m  ( 1 ... 1 ) ) )
691, 7nnaddcld 10581 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  +  W
)  e.  NN )
7069adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  W )  e.  NN )
71 elfznn0 11769 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  m  e.  NN0 )
723nnnn0d 10851 . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  NN0 )
73 nn0mulcl 10831 . . . . . . . . . . . . . 14  |-  ( ( m  e.  NN0  /\  D  e.  NN0 )  -> 
( m  x.  D
)  e.  NN0 )
7471, 72, 73syl2anr 478 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
m  x.  D )  e.  NN0 )
75 nnnn0addcl 10825 . . . . . . . . . . . . 13  |-  ( ( ( A  +  W
)  e.  NN  /\  ( m  x.  D
)  e.  NN0 )  ->  ( ( A  +  W )  +  ( m  x.  D ) )  e.  NN )
7670, 74, 75syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  e.  NN )
77 nnuz 11116 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
7876, 77syl6eleq 2565 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  e.  ( ZZ>= `  1
) )
7916adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A (AP `  K ) D )  C_  ( `' G " { C } ) )
80 eqid 2467 . . . . . . . . . . . . . . . . . 18  |-  ( A  +  ( m  x.  D ) )  =  ( A  +  ( m  x.  D ) )
81 oveq1 6290 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  m  ->  (
n  x.  D )  =  ( m  x.  D ) )
8281oveq2d 6299 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  m  ->  ( A  +  ( n  x.  D ) )  =  ( A  +  ( m  x.  D ) ) )
8382eqeq2d 2481 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  m  ->  (
( A  +  ( m  x.  D ) )  =  ( A  +  ( n  x.  D ) )  <->  ( A  +  ( m  x.  D ) )  =  ( A  +  ( m  x.  D ) ) ) )
8483rspcev 3214 . . . . . . . . . . . . . . . . . 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 671 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  E. n  e.  ( 0 ... ( K  -  1 ) ) ( A  +  ( m  x.  D
) )  =  ( A  +  ( n  x.  D ) ) )
8633nnnn0d 10851 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  K  e.  NN0 )
87 vdwapval 14349 . . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  ( m  x.  D ) )  e.  ( A (AP `  K ) D ) )
9179, 90sseldd 3505 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  ( m  x.  D ) )  e.  ( `' G " { C } ) )
9218, 19fnmpti 5708 . . . . . . . . . . . . . . . 16  |-  G  Fn  ( 1 ... W
)
93 fniniseg 6001 . . . . . . . . . . . . . . . 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 5 . . . . . . . . . . . . . . 15  |-  ( ( A  +  ( m  x.  D ) )  e.  ( `' G " { C } )  <-> 
( ( A  +  ( m  x.  D
) )  e.  ( 1 ... W )  /\  ( G `  ( A  +  (
m  x.  D ) ) )  =  C ) )
9591, 94sylib 196 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  ( m  x.  D ) )  e.  ( 1 ... W )  /\  ( G `  ( A  +  ( m  x.  D ) ) )  =  C ) )
9695simpld 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( A  +  ( m  x.  D ) )  e.  ( 1 ... W
) )
97 elfzuz3 11684 . . . . . . . . . . . . 13  |-  ( ( A  +  ( m  x.  D ) )  e.  ( 1 ... W )  ->  W  e.  ( ZZ>= `  ( A  +  ( m  x.  D ) ) ) )
98 eluzelz 11090 . . . . . . . . . . . . . 14  |-  ( W  e.  ( ZZ>= `  ( A  +  ( m  x.  D ) ) )  ->  W  e.  ZZ )
99 eluzadd 11109 . . . . . . . . . . . . . 14  |-  ( ( W  e.  ( ZZ>= `  ( A  +  (
m  x.  D ) ) )  /\  W  e.  ZZ )  ->  ( W  +  W )  e.  ( ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
10098, 99mpdan 668 . . . . . . . . . . . . 13  |-  ( W  e.  ( ZZ>= `  ( A  +  ( m  x.  D ) ) )  ->  ( W  +  W )  e.  (
ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
10196, 97, 1003syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( W  +  W )  e.  ( ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
10282timesd 10780 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  W
)  =  ( W  +  W ) )
103102adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
2  x.  W )  =  ( W  +  W ) )
1042adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  A  e.  CC )
1058adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  W  e.  CC )
10674nn0cnd 10853 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
m  x.  D )  e.  CC )
107104, 105, 106add32d 9801 . . . . . . . . . . . . 13  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  =  ( ( A  +  ( m  x.  D ) )  +  W ) )
108107fveq2d 5869 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( ZZ>=
`  ( ( A  +  W )  +  ( m  x.  D
) ) )  =  ( ZZ>= `  ( ( A  +  ( m  x.  D ) )  +  W ) ) )
109101, 103, 1083eltr4d 2570 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
2  x.  W )  e.  ( ZZ>= `  (
( A  +  W
)  +  ( m  x.  D ) ) ) )
110 elfzuzb 11681 . . . . . . . . . . 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 664 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
( A  +  W
)  +  ( m  x.  D ) )  e.  ( 1 ... ( 2  x.  W
) ) )
112107fveq2d 5869 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) )  =  ( F `  ( ( A  +  ( m  x.  D
) )  +  W
) ) )
113 oveq1 6290 . . . . . . . . . . . . . 14  |-  ( x  =  ( A  +  ( m  x.  D
) )  ->  (
x  +  W )  =  ( ( A  +  ( m  x.  D ) )  +  W ) )
114113fveq2d 5869 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +  ( m  x.  D
) )  ->  ( F `  ( x  +  W ) )  =  ( F `  (
( A  +  ( m  x.  D ) )  +  W ) ) )
115 fvex 5875 . . . . . . . . . . . . 13  |-  ( F `
 ( ( A  +  ( m  x.  D ) )  +  W ) )  e. 
_V
116114, 19, 115fvmpt 5949 . . . . . . . . . . . 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 463 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( G `  ( A  +  ( m  x.  D ) ) )  =  C )
119112, 117, 1183eqtr2d 2514 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) )  =  C )
120111, 119jca 532 . . . . . . . . 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 2539 . . . . . . . . . 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 5865 . . . . . . . . . . 11  |-  ( x  =  ( ( A  +  W )  +  ( m  x.  D
) )  ->  ( F `  x )  =  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) ) )
123122eqeq1d 2469 . . . . . . . . . 10  |-  ( x  =  ( ( A  +  W )  +  ( m  x.  D
) )  ->  (
( F `  x
)  =  C  <->  ( F `  ( ( A  +  W )  +  ( m  x.  D ) ) )  =  C ) )
124121, 123anbi12d 710 . . . . . . . . 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 222 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  (
x  =  ( ( A  +  W )  +  ( m  x.  D ) )  -> 
( x  e.  ( 1 ... ( 2  x.  W ) )  /\  ( F `  x )  =  C ) ) )
126125rexlimdva 2955 . . . . . . 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 14349 . . . . . . . 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 1228 . . . . . . 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 5730 . . . . . . . 8  |-  ( F : ( 1 ... ( 2  x.  W
) ) --> R  ->  F  Fn  ( 1 ... ( 2  x.  W ) ) )
131 fniniseg 6001 . . . . . . . 8  |-  ( F  Fn  ( 1 ... ( 2  x.  W
) )  ->  (
x  e.  ( `' F " { C } )  <->  ( x  e.  ( 1 ... (
2  x.  W ) )  /\  ( F `
 x )  =  C ) ) )
132129, 130, 1313syl 20 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( `' F " { C } )  <->  ( x  e.  ( 1 ... (
2  x.  W ) )  /\  ( F `
 x )  =  C ) ) )
133126, 128, 1323imtr4d 268 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( A  +  W
) (AP `  K
) D )  ->  x  e.  ( `' F " { C }
) ) )
134133ssrdv 3510 . . . . 5  |-  ( ph  ->  ( ( A  +  W ) (AP `  K ) D ) 
C_  ( `' F " { C } ) )
135 fvsng 6094 . . . . . . . . 9  |-  ( ( 1  e.  NN  /\  D  e.  NN )  ->  ( { <. 1 ,  D >. } `  1
)  =  D )
13652, 3, 135sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( { <. 1 ,  D >. } `  1
)  =  D )
137136oveq2d 6299 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) )  =  ( ( A  +  ( W  -  D ) )  +  D ) )
1382, 12, 4addassd 9617 . . . . . . 7  |-  ( ph  ->  ( ( A  +  ( W  -  D
) )  +  D
)  =  ( A  +  ( ( W  -  D )  +  D ) ) )
1398, 4npcand 9933 . . . . . . . 8  |-  ( ph  ->  ( ( W  -  D )  +  D
)  =  W )
140139oveq2d 6299 . . . . . . 7  |-  ( ph  ->  ( A  +  ( ( W  -  D
)  +  D ) )  =  ( A  +  W ) )
141137, 138, 1403eqtrd 2512 . . . . . 6  |-  ( ph  ->  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) )  =  ( A  +  W ) )
142141, 136oveq12d 6301 . . . . 5  |-  ( ph  ->  ( ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) (AP `  K ) ( {
<. 1 ,  D >. } `  1 ) )  =  ( ( A  +  W ) (AP `  K ) D ) )
143141fveq2d 5869 . . . . . . . 8  |-  ( ph  ->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) )  =  ( F `  ( A  +  W ) ) )
144 vdwapid1 14351 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  A  e.  NN  /\  D  e.  NN )  ->  A  e.  ( A (AP `  K ) D ) )
14533, 1, 3, 144syl3anc 1228 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ( A (AP `  K ) D ) )
14616, 145sseldd 3505 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( `' G " { C } ) )
147 fniniseg 6001 . . . . . . . . . . . 12  |-  ( G  Fn  ( 1 ... W )  ->  ( A  e.  ( `' G " { C }
)  <->  ( A  e.  ( 1 ... W
)  /\  ( G `  A )  =  C ) ) )
14892, 147ax-mp 5 . . . . . . . . . . 11  |-  ( A  e.  ( `' G " { C } )  <-> 
( A  e.  ( 1 ... W )  /\  ( G `  A )  =  C ) )
149146, 148sylib 196 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( 1 ... W )  /\  ( G `  A )  =  C ) )
150149simpld 459 . . . . . . . . 9  |-  ( ph  ->  A  e.  ( 1 ... W ) )
151 oveq1 6290 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x  +  W )  =  ( A  +  W ) )
152151fveq2d 5869 . . . . . . . . . 10  |-  ( x  =  A  ->  ( F `  ( x  +  W ) )  =  ( F `  ( A  +  W )
) )
153 fvex 5875 . . . . . . . . . 10  |-  ( F `
 ( A  +  W ) )  e. 
_V
154152, 19, 153fvmpt 5949 . . . . . . . . 9  |-  ( A  e.  ( 1 ... W )  ->  ( G `  A )  =  ( F `  ( A  +  W
) ) )
155150, 154syl 16 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  =  ( F `
 ( A  +  W ) ) )
156149simprd 463 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  =  C )
157143, 155, 1563eqtr2d 2514 . . . . . . 7  |-  ( ph  ->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) )  =  C )
158157sneqd 4039 . . . . . 6  |-  ( ph  ->  { ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) }  =  { C } )
159158imaeq2d 5336 . . . . 5  |-  ( ph  ->  ( `' F " { ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) } )  =  ( `' F " { C } ) )
160134, 142, 1593sstr4d 3547 . . . 4  |-  ( ph  ->  ( ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) (AP `  K ) ( {
<. 1 ,  D >. } `  1 ) )  C_  ( `' F " { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) ) } ) )
161160ralrimivw 2879 . . 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 4534 . . . . . . . 8  |-  ( ph  ->  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) )  =  ( i  e.  ( 1 ... 1 ) 
|->  C ) )
163 fconstmpt 5042 . . . . . . . 8  |-  ( ( 1 ... 1 )  X.  { C }
)  =  ( i  e.  ( 1 ... 1 )  |->  C )
164162, 163syl6eqr 2526 . . . . . . 7  |-  ( ph  ->  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) )  =  ( ( 1 ... 1 )  X.  { C } ) )
165164rneqd 5229 . . . . . 6  |-  ( ph  ->  ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) )  =  ran  ( ( 1 ... 1 )  X. 
{ C } ) )
166 elfz3 11695 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  ( 1 ... 1
) )
167 ne0i 3791 . . . . . . . 8  |-  ( 1  e.  ( 1 ... 1 )  ->  (
1 ... 1 )  =/=  (/) )
16860, 166, 167mp2b 10 . . . . . . 7  |-  ( 1 ... 1 )  =/=  (/)
169 rnxp 5436 . . . . . . 7  |-  ( ( 1 ... 1 )  =/=  (/)  ->  ran  ( ( 1 ... 1 )  X.  { C }
)  =  { C } )
170168, 169ax-mp 5 . . . . . 6  |-  ran  (
( 1 ... 1
)  X.  { C } )  =  { C }
171165, 170syl6eq 2524 . . . . 5  |-  ( ph  ->  ran  ( i  e.  ( 1 ... 1
)  |->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( { <. 1 ,  D >. } `  1 ) ) ) )  =  { C } )
172171fveq2d 5869 . . . 4  |-  ( ph  ->  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) ) )  =  ( # `  { C } ) )
173 vdwlem8.c . . . . 5  |-  C  e. 
_V
174 hashsng 12405 . . . . 5  |-  ( C  e.  _V  ->  ( # `
 { C }
)  =  1 )
175173, 174ax-mp 5 . . . 4  |-  ( # `  { C } )  =  1
176172, 175syl6eq 2524 . . 3  |-  ( ph  ->  ( # `  ran  ( i  e.  ( 1 ... 1 ) 
|->  ( F `  (
( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) ) ) )  =  1 )
177 oveq1 6290 . . . . . . . 8  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
a  +  ( d `
 i ) )  =  ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) )
178177oveq1d 6298 . . . . . . 7  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  =  ( ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) )
179177fveq2d 5869 . . . . . . . . 9  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  ( F `  ( a  +  ( d `  i ) ) )  =  ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) ) )
180179sneqd 4039 . . . . . . . 8  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  { ( F `  ( a  +  ( d `  i ) ) ) }  =  { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) } )
181180imaeq2d 5336 . . . . . . 7  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  ( `' F " { ( F `  ( a  +  ( d `  i ) ) ) } )  =  ( `' F " { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i ) ) ) } ) )
182178, 181sseq12d 3533 . . . . . 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 2903 . . . . 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 4534 . . . . . . . 8  |-  ( a  =  ( A  +  ( W  -  D
) )  ->  (
i  e.  ( 1 ... 1 )  |->  ( F `  ( a  +  ( d `  i ) ) ) )  =  ( i  e.  ( 1 ... 1 )  |->  ( F `
 ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) ) ) )
185184rneqd 5229 . . . . . . 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 5869 . . . . . 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 2469 . . . . 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 710 . . . 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 5864 . . . . . . . . . 10  |-  ( d  =  { <. 1 ,  D >. }  ->  (
d `  i )  =  ( { <. 1 ,  D >. } `
 i ) )
190 elfz1eq 11696 . . . . . . . . . . 11  |-  ( i  e.  ( 1 ... 1 )  ->  i  =  1 )
191190fveq2d 5869 . . . . . . . . . 10  |-  ( i  e.  ( 1 ... 1 )  ->  ( { <. 1 ,  D >. } `  i )  =  ( { <. 1 ,  D >. } `
 1 ) )
192189, 191sylan9eq 2528 . . . . . . . . 9  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( d `  i )  =  ( { <. 1 ,  D >. } `  1 ) )
193192oveq2d 6299 . . . . . . . 8  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( ( A  +  ( W  -  D ) )  +  ( d `  i
) )  =  ( ( A  +  ( W  -  D ) )  +  ( {
<. 1 ,  D >. } `  1 ) ) )
194193, 192oveq12d 6301 . . . . . . 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 5869 . . . . . . . . 9  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  ( F `  ( ( A  +  ( W  -  D
) )  +  ( d `  i ) ) )  =  ( F `  ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `
 1 ) ) ) )
196195sneqd 4039 . . . . . . . 8  |-  ( ( d  =  { <. 1 ,  D >. }  /\  i  e.  ( 1 ... 1 ) )  ->  { ( F `  ( ( A  +  ( W  -  D ) )  +  ( d `  i
) ) ) }  =  { ( F `
 ( ( A  +  ( W  -  D ) )  +  ( { <. 1 ,  D >. } `  1
) ) ) } )
197196imaeq2d 5336 . . . . . . 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 3533 . . . . . 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 2900 . . . . 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 4533 . . . . . . . 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 5229 . . . . . . 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 5869 . . . . . 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 2469 . . . . 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 710 . . . 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 3225 . . 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 1232 . 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 6308 . . 3  |-  ( 1 ... ( 2  x.  W ) )  e. 
_V
20852a1i 11 . . 3  |-  ( ph  ->  1  e.  NN )
209 eqid 2467 . . 3  |-  ( 1 ... 1 )  =  ( 1 ... 1
)
210207, 86, 129, 208, 209vdwpc 14356 . 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 232 1  |-  ( ph  -> 
<. 1 ,  K >. PolyAP 
F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    Fn wfn 5582   -->wf 5583   -1-1-onto->wf1o 5586   ` cfv 5587  (class class class)co 6283    ^m cmap 7420   Fincfn 7516   CCcc 9489   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496    < clt 9627    - cmin 9804   NNcn 10535   2c2 10584   NN0cn0 10794   ZZcz 10863   ZZ>=cuz 11081   ...cfz 11671   #chash 12372  APcvdwa 14341   PolyAP cvdwp 14343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-hash 12373  df-vdwap 14344  df-vdwpc 14346
This theorem is referenced by:  vdwlem10  14366
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