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Theorem vdwlem7 14053
Description: Lemma for vdw 14060. (Contributed by Mario Carneiro, 12-Sep-2014.)
Hypotheses
Ref Expression
vdwlem3.v  |-  ( ph  ->  V  e.  NN )
vdwlem3.w  |-  ( ph  ->  W  e.  NN )
vdwlem4.r  |-  ( ph  ->  R  e.  Fin )
vdwlem4.h  |-  ( ph  ->  H : ( 1 ... ( W  x.  ( 2  x.  V
) ) ) --> R )
vdwlem4.f  |-  F  =  ( x  e.  ( 1 ... V ) 
|->  ( y  e.  ( 1 ... W ) 
|->  ( H `  (
y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )
vdwlem7.m  |-  ( ph  ->  M  e.  NN )
vdwlem7.g  |-  ( ph  ->  G : ( 1 ... W ) --> R )
vdwlem7.k  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
vdwlem7.a  |-  ( ph  ->  A  e.  NN )
vdwlem7.d  |-  ( ph  ->  D  e.  NN )
vdwlem7.s  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' F " { G } ) )
Assertion
Ref Expression
vdwlem7  |-  ( ph  ->  ( <. M ,  K >. PolyAP 
G  ->  ( <. ( M  +  1 ) ,  K >. PolyAP  H  \/  ( K  +  1
) MonoAP  G ) ) )
Distinct variable groups:    x, y, A    x, G, y    x, K, y    ph, x, y   
x, R, y    x, H, y    x, M, y   
x, D, y    x, W, y    x, V, y
Allowed substitution hints:    F( x, y)

Proof of Theorem vdwlem7
Dummy variables  k 
a  d  i  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6121 . . 3  |-  ( 1 ... W )  e. 
_V
2 2nn0 10601 . . . 4  |-  2  e.  NN0
3 vdwlem7.k . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
4 eluznn0 10929 . . . 4  |-  ( ( 2  e.  NN0  /\  K  e.  ( ZZ>= ` 
2 ) )  ->  K  e.  NN0 )
52, 3, 4sylancr 663 . . 3  |-  ( ph  ->  K  e.  NN0 )
6 vdwlem7.g . . 3  |-  ( ph  ->  G : ( 1 ... W ) --> R )
7 vdwlem7.m . . 3  |-  ( ph  ->  M  e.  NN )
8 eqid 2443 . . 3  |-  ( 1 ... M )  =  ( 1 ... M
)
91, 5, 6, 7, 8vdwpc 14046 . 2  |-  ( ph  ->  ( <. M ,  K >. PolyAP 
G  <->  E. a  e.  NN  E. d  e.  ( NN 
^m  ( 1 ... M ) ) ( A. i  e.  ( 1 ... M ) ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' G " { ( G `  ( a  +  ( d `  i ) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M ) 
|->  ( G `  (
a  +  ( d `
 i ) ) ) ) )  =  M ) ) )
10 vdwlem3.v . . . . . 6  |-  ( ph  ->  V  e.  NN )
1110ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  V  e.  NN )
12 vdwlem3.w . . . . . 6  |-  ( ph  ->  W  e.  NN )
1312ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  W  e.  NN )
14 vdwlem4.r . . . . . 6  |-  ( ph  ->  R  e.  Fin )
1514ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  R  e.  Fin )
16 vdwlem4.h . . . . . 6  |-  ( ph  ->  H : ( 1 ... ( W  x.  ( 2  x.  V
) ) ) --> R )
1716ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  H : ( 1 ... ( W  x.  ( 2  x.  V ) ) ) --> R )
18 vdwlem4.f . . . . 5  |-  F  =  ( x  e.  ( 1 ... V ) 
|->  ( y  e.  ( 1 ... W ) 
|->  ( H `  (
y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )
197ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  M  e.  NN )
206ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  G : ( 1 ... W ) --> R )
213ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  K  e.  (
ZZ>= `  2 ) )
22 vdwlem7.a . . . . . 6  |-  ( ph  ->  A  e.  NN )
2322ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  A  e.  NN )
24 vdwlem7.d . . . . . 6  |-  ( ph  ->  D  e.  NN )
2524ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  D  e.  NN )
26 vdwlem7.s . . . . . 6  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' F " { G } ) )
2726ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  ( A (AP
`  K ) D )  C_  ( `' F " { G }
) )
28 simplrl 759 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  a  e.  NN )
29 simplrr 760 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  d  e.  ( NN  ^m  ( 1 ... M ) ) )
30 nnex 10333 . . . . . . 7  |-  NN  e.  _V
31 ovex 6121 . . . . . . 7  |-  ( 1 ... M )  e. 
_V
3230, 31elmap 7246 . . . . . 6  |-  ( d  e.  ( NN  ^m  ( 1 ... M
) )  <->  d :
( 1 ... M
) --> NN )
3329, 32sylib 196 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  d : ( 1 ... M ) --> NN )
34 simprl 755 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  A. i  e.  ( 1 ... M ) ( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' G " { ( G `  ( a  +  ( d `  i ) ) ) } ) )
35 fveq2 5696 . . . . . . . . . 10  |-  ( i  =  k  ->  (
d `  i )  =  ( d `  k ) )
3635oveq2d 6112 . . . . . . . . 9  |-  ( i  =  k  ->  (
a  +  ( d `
 i ) )  =  ( a  +  ( d `  k
) ) )
3736, 35oveq12d 6114 . . . . . . . 8  |-  ( i  =  k  ->  (
( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  =  ( ( a  +  ( d `  k
) ) (AP `  K ) ( d `
 k ) ) )
3836fveq2d 5700 . . . . . . . . . 10  |-  ( i  =  k  ->  ( G `  ( a  +  ( d `  i ) ) )  =  ( G `  ( a  +  ( d `  k ) ) ) )
3938sneqd 3894 . . . . . . . . 9  |-  ( i  =  k  ->  { ( G `  ( a  +  ( d `  i ) ) ) }  =  { ( G `  ( a  +  ( d `  k ) ) ) } )
4039imaeq2d 5174 . . . . . . . 8  |-  ( i  =  k  ->  ( `' G " { ( G `  ( a  +  ( d `  i ) ) ) } )  =  ( `' G " { ( G `  ( a  +  ( d `  k ) ) ) } ) )
4137, 40sseq12d 3390 . . . . . . 7  |-  ( i  =  k  ->  (
( ( a  +  ( d `  i
) ) (AP `  K ) ( d `
 i ) ) 
C_  ( `' G " { ( G `  ( a  +  ( d `  i ) ) ) } )  <-> 
( ( a  +  ( d `  k
) ) (AP `  K ) ( d `
 k ) ) 
C_  ( `' G " { ( G `  ( a  +  ( d `  k ) ) ) } ) ) )
4241cbvralv 2952 . . . . . 6  |-  ( A. i  e.  ( 1 ... M ) ( ( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  C_  ( `' G " { ( G `  ( a  +  ( d `  i ) ) ) } )  <->  A. k  e.  ( 1 ... M
) ( ( a  +  ( d `  k ) ) (AP
`  K ) ( d `  k ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  k
) ) ) } ) )
4334, 42sylib 196 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  A. k  e.  ( 1 ... M ) ( ( a  +  ( d `  k
) ) (AP `  K ) ( d `
 k ) ) 
C_  ( `' G " { ( G `  ( a  +  ( d `  k ) ) ) } ) )
4438cbvmptv 4388 . . . . 5  |-  ( i  e.  ( 1 ... M )  |->  ( G `
 ( a  +  ( d `  i
) ) ) )  =  ( k  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  k ) ) ) )
45 simprr 756 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  ( # `  ran  ( i  e.  ( 1 ... M ) 
|->  ( G `  (
a  +  ( d `
 i ) ) ) ) )  =  M )
46 eqid 2443 . . . . 5  |-  ( a  +  ( W  x.  ( ( A  +  ( V  -  D
) )  -  1 ) ) )  =  ( a  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )
47 eqid 2443 . . . . 5  |-  ( j  e.  ( 1 ... ( M  +  1 ) )  |->  ( if ( j  =  ( M  +  1 ) ,  0 ,  ( d `  j ) )  +  ( W  x.  D ) ) )  =  ( j  e.  ( 1 ... ( M  +  1 ) )  |->  ( if ( j  =  ( M  +  1 ) ,  0 ,  ( d `  j ) )  +  ( W  x.  D ) ) )
4811, 13, 15, 17, 18, 19, 20, 21, 23, 25, 27, 28, 33, 43, 44, 45, 46, 47vdwlem6 14052 . . . 4  |-  ( ( ( ph  /\  (
a  e.  NN  /\  d  e.  ( NN  ^m  ( 1 ... M
) ) ) )  /\  ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M ) )  ->  ( <. ( M  +  1 ) ,  K >. PolyAP  H  \/  ( K  +  1
) MonoAP  G ) )
4948ex 434 . . 3  |-  ( (
ph  /\  ( a  e.  NN  /\  d  e.  ( NN  ^m  (
1 ... M ) ) ) )  ->  (
( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M )  -> 
( <. ( M  + 
1 ) ,  K >. PolyAP 
H  \/  ( K  +  1 ) MonoAP  G
) ) )
5049rexlimdvva 2853 . 2  |-  ( ph  ->  ( E. a  e.  NN  E. d  e.  ( NN  ^m  (
1 ... M ) ) ( A. i  e.  ( 1 ... M
) ( ( a  +  ( d `  i ) ) (AP
`  K ) ( d `  i ) )  C_  ( `' G " { ( G `
 ( a  +  ( d `  i
) ) ) } )  /\  ( # `  ran  ( i  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  i ) ) ) ) )  =  M )  -> 
( <. ( M  + 
1 ) ,  K >. PolyAP 
H  \/  ( K  +  1 ) MonoAP  G
) ) )
519, 50sylbid 215 1  |-  ( ph  ->  ( <. M ,  K >. PolyAP 
G  ->  ( <. ( M  +  1 ) ,  K >. PolyAP  H  \/  ( K  +  1
) MonoAP  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    C_ wss 3333   ifcif 3796   {csn 3882   <.cop 3888   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   ran crn 4846   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   Fincfn 7315   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    - cmin 9600   NNcn 10327   2c2 10376   NN0cn0 10584   ZZ>=cuz 10866   ...cfz 11442   #chash 12108  APcvdwa 14031   MonoAP cvdwm 14032   PolyAP cvdwp 14033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-hash 12109  df-vdwap 14034  df-vdwmc 14035  df-vdwpc 14036
This theorem is referenced by:  vdwlem9  14055
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