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Theorem vdwlem7 14873
Description: Lemma for vdw 14880. (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 6270 . . 3  |-  ( 1 ... W )  e. 
_V
2 2nn0 10830 . . . 4  |-  2  e.  NN0
3 vdwlem7.k . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
4 eluznn0 11172 . . . 4  |-  ( ( 2  e.  NN0  /\  K  e.  ( ZZ>= ` 
2 ) )  ->  K  e.  NN0 )
52, 3, 4sylancr 667 . . 3  |-  ( ph  ->  K  e.  NN0 )
6 vdwlem7.g . . 3  |-  ( ph  ->  G : ( 1 ... W ) --> R )
7 vdwlem7.m . . 3  |-  ( ph  ->  M  e.  NN )
8 eqid 2422 . . 3  |-  ( 1 ... M )  =  ( 1 ... M
)
91, 5, 6, 7, 8vdwpc 14866 . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 730 . . . . 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 768 . . . . 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 769 . . . . . 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 10559 . . . . . . 7  |-  NN  e.  _V
31 ovex 6270 . . . . . . 7  |-  ( 1 ... M )  e. 
_V
3230, 31elmap 7448 . . . . . 6  |-  ( d  e.  ( NN  ^m  ( 1 ... M
) )  <->  d :
( 1 ... M
) --> NN )
3329, 32sylib 199 . . . . 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 762 . . . . . 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 5818 . . . . . . . . . 10  |-  ( i  =  k  ->  (
d `  i )  =  ( d `  k ) )
3635oveq2d 6258 . . . . . . . . 9  |-  ( i  =  k  ->  (
a  +  ( d `
 i ) )  =  ( a  +  ( d `  k
) ) )
3736, 35oveq12d 6260 . . . . . . . 8  |-  ( i  =  k  ->  (
( a  +  ( d `  i ) ) (AP `  K
) ( d `  i ) )  =  ( ( a  +  ( d `  k
) ) (AP `  K ) ( d `
 k ) ) )
3836fveq2d 5822 . . . . . . . . . 10  |-  ( i  =  k  ->  ( G `  ( a  +  ( d `  i ) ) )  =  ( G `  ( a  +  ( d `  k ) ) ) )
3938sneqd 3946 . . . . . . . . 9  |-  ( i  =  k  ->  { ( G `  ( a  +  ( d `  i ) ) ) }  =  { ( G `  ( a  +  ( d `  k ) ) ) } )
4039imaeq2d 5123 . . . . . . . 8  |-  ( i  =  k  ->  ( `' G " { ( G `  ( a  +  ( d `  i ) ) ) } )  =  ( `' G " { ( G `  ( a  +  ( d `  k ) ) ) } ) )
4137, 40sseq12d 3429 . . . . . . 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 2990 . . . . . 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 199 . . . . 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 4452 . . . . 5  |-  ( i  e.  ( 1 ... M )  |->  ( G `
 ( a  +  ( d `  i
) ) ) )  =  ( k  e.  ( 1 ... M
)  |->  ( G `  ( a  +  ( d `  k ) ) ) )
45 simprr 764 . . . . 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 2422 . . . . 5  |-  ( a  +  ( W  x.  ( ( A  +  ( V  -  D
) )  -  1 ) ) )  =  ( a  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )
47 eqid 2422 . . . . 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 14872 . . . 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 435 . . 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 2857 . 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 218 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 369    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2708   E.wrex 2709    C_ wss 3372   ifcif 3847   {csn 3934   <.cop 3940   class class class wbr 4359    |-> cmpt 4418   `'ccnv 4788   ran crn 4790   "cima 4792   -->wf 5533   ` cfv 5537  (class class class)co 6242    ^m cmap 7420   Fincfn 7517   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    - cmin 9804   NNcn 10553   2c2 10603   NN0cn0 10813   ZZ>=cuz 11103   ...cfz 11728   #chash 12458  APcvdwa 14851   MonoAP cvdwm 14852   PolyAP cvdwp 14853
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 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rmo 2716  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-int 4192  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-om 6644  df-1st 6744  df-2nd 6745  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8318  df-cda 8542  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-nn 10554  df-2 10612  df-n0 10814  df-z 10882  df-uz 11104  df-rp 11247  df-fz 11729  df-hash 12459  df-vdwap 14854  df-vdwmc 14855  df-vdwpc 14856
This theorem is referenced by:  vdwlem9  14875
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