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Theorem vdwlem5 14038
Description: Lemma for vdw 14047. (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 } ) )
vdwlem6.b  |-  ( ph  ->  B  e.  NN )
vdwlem6.e  |-  ( ph  ->  E : ( 1 ... M ) --> NN )
vdwlem6.s  |-  ( ph  ->  A. i  e.  ( 1 ... M ) ( ( B  +  ( E `  i ) ) (AP `  K
) ( E `  i ) )  C_  ( `' G " { ( G `  ( B  +  ( E `  i ) ) ) } ) )
vdwlem6.j  |-  J  =  ( i  e.  ( 1 ... M ) 
|->  ( G `  ( B  +  ( E `  i ) ) ) )
vdwlem6.r  |-  ( ph  ->  ( # `  ran  J )  =  M )
vdwlem6.t  |-  T  =  ( B  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )
vdwlem6.p  |-  P  =  ( j  e.  ( 1 ... ( M  +  1 ) ) 
|->  ( if ( j  =  ( M  + 
1 ) ,  0 ,  ( E `  j ) )  +  ( W  x.  D
) ) )
Assertion
Ref Expression
vdwlem5  |-  ( ph  ->  T  e.  NN )
Distinct variable groups:    x, y, A    i, j, x, y, G    i, K, j, x, y    i, J, j, x    P, i, x    ph, i, j, x, y    R, i, x, y    B, i, j, x, y   
i, H, x, y   
i, M, j, x, y    D, j, x, y   
i, E, j, x, y    i, W, j, x, y    T, i, x    x, V, y
Allowed substitution hints:    A( i, j)    D( i)    P( y, j)    R( j)    T( y, j)    F( x, y, i, j)    H( j)    J( y)    V( i, j)

Proof of Theorem vdwlem5
StepHypRef Expression
1 vdwlem6.t . 2  |-  T  =  ( B  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )
2 vdwlem6.b . . 3  |-  ( ph  ->  B  e.  NN )
3 vdwlem3.w . . . . 5  |-  ( ph  ->  W  e.  NN )
43nnnn0d 10628 . . . 4  |-  ( ph  ->  W  e.  NN0 )
5 vdwlem7.a . . . . . 6  |-  ( ph  ->  A  e.  NN )
6 vdwlem3.v . . . . . . . . . 10  |-  ( ph  ->  V  e.  NN )
76nncnd 10330 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
8 vdwlem7.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  NN )
98nncnd 10330 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
107, 9subcld 9711 . . . . . . . 8  |-  ( ph  ->  ( V  -  D
)  e.  CC )
115nncnd 10330 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1210, 11npcand 9715 . . . . . . 7  |-  ( ph  ->  ( ( ( V  -  D )  -  A )  +  A
)  =  ( V  -  D ) )
137, 9, 11subsub4d 9742 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  -  D )  -  A
)  =  ( V  -  ( D  +  A ) ) )
149, 11addcomd 9563 . . . . . . . . . . 11  |-  ( ph  ->  ( D  +  A
)  =  ( A  +  D ) )
1514oveq2d 6102 . . . . . . . . . 10  |-  ( ph  ->  ( V  -  ( D  +  A )
)  =  ( V  -  ( A  +  D ) ) )
1613, 15eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( ( V  -  D )  -  A
)  =  ( V  -  ( A  +  D ) ) )
17 cnvimass 5184 . . . . . . . . . . . . 13  |-  ( `' F " { G } )  C_  dom  F
18 vdwlem4.r . . . . . . . . . . . . . . 15  |-  ( ph  ->  R  e.  Fin )
19 vdwlem4.h . . . . . . . . . . . . . . 15  |-  ( ph  ->  H : ( 1 ... ( W  x.  ( 2  x.  V
) ) ) --> R )
20 vdwlem4.f . . . . . . . . . . . . . . 15  |-  F  =  ( x  e.  ( 1 ... V ) 
|->  ( y  e.  ( 1 ... W ) 
|->  ( H `  (
y  +  ( W  x.  ( ( x  -  1 )  +  V ) ) ) ) ) )
216, 3, 18, 19, 20vdwlem4 14037 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : ( 1 ... V ) --> ( R  ^m  ( 1 ... W ) ) )
22 fdm 5558 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... V ) --> ( R  ^m  ( 1 ... W ) )  ->  dom  F  =  ( 1 ... V ) )
2321, 22syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  =  ( 1 ... V ) )
2417, 23syl5sseq 3399 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F " { G } )  C_  ( 1 ... V
) )
25 vdwlem7.s . . . . . . . . . . . . 13  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' F " { G } ) )
26 ssun2 3515 . . . . . . . . . . . . . . 15  |-  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D )  C_  ( { A }  u.  (
( A  +  D
) (AP `  ( K  -  1 ) ) D ) )
27 vdwlem7.k . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
28 uz2m1nn 10921 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ( ZZ>= `  2
)  ->  ( K  -  1 )  e.  NN )
2927, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  -  1 )  e.  NN )
305, 8nnaddcld 10360 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  +  D
)  e.  NN )
31 vdwapid1 14028 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  -  1 )  e.  NN  /\  ( A  +  D
)  e.  NN  /\  D  e.  NN )  ->  ( A  +  D
)  e.  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D ) )
3229, 30, 8, 31syl3anc 1218 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  +  D
)  e.  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D ) )
3326, 32sseldi 3349 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  +  D
)  e.  ( { A }  u.  (
( A  +  D
) (AP `  ( K  -  1 ) ) D ) ) )
34 eluz2b2 10919 . . . . . . . . . . . . . . . . . . . . 21  |-  ( K  e.  ( ZZ>= `  2
)  <->  ( K  e.  NN  /\  1  < 
K ) )
3534simplbi 460 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  NN )
3627, 35syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  K  e.  NN )
3736nncnd 10330 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  K  e.  CC )
38 ax-1cn 9332 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
39 npcan 9611 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
4037, 38, 39sylancl 662 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
4140fveq2d 5690 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (AP `  ( ( K  -  1 )  +  1 ) )  =  (AP `  K
) )
4241oveqd 6103 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( A (AP
`  K ) D ) )
43 nnm1nn0 10613 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
4436, 43syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
45 vdwapun 14027 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  -  1 )  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( A (AP `  ( ( K  -  1 )  +  1 ) ) D )  =  ( { A }  u.  ( ( A  +  D ) (AP `  ( K  -  1
) ) D ) ) )
4644, 5, 8, 45syl3anc 1218 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4742, 46eqtr3d 2472 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A (AP `  K ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4833, 47eleqtrrd 2515 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  D
)  e.  ( A (AP `  K ) D ) )
4925, 48sseldd 3352 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  D
)  e.  ( `' F " { G } ) )
5024, 49sseldd 3352 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  D
)  e.  ( 1 ... V ) )
51 elfzuz3 11442 . . . . . . . . . . 11  |-  ( ( A  +  D )  e.  ( 1 ... V )  ->  V  e.  ( ZZ>= `  ( A  +  D ) ) )
5250, 51syl 16 . . . . . . . . . 10  |-  ( ph  ->  V  e.  ( ZZ>= `  ( A  +  D
) ) )
53 uznn0sub 10884 . . . . . . . . . 10  |-  ( V  e.  ( ZZ>= `  ( A  +  D )
)  ->  ( V  -  ( A  +  D ) )  e. 
NN0 )
5452, 53syl 16 . . . . . . . . 9  |-  ( ph  ->  ( V  -  ( A  +  D )
)  e.  NN0 )
5516, 54eqeltrd 2512 . . . . . . . 8  |-  ( ph  ->  ( ( V  -  D )  -  A
)  e.  NN0 )
56 nn0nnaddcl 10603 . . . . . . . 8  |-  ( ( ( ( V  -  D )  -  A
)  e.  NN0  /\  A  e.  NN )  ->  ( ( ( V  -  D )  -  A )  +  A
)  e.  NN )
5755, 5, 56syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( ( V  -  D )  -  A )  +  A
)  e.  NN )
5812, 57eqeltrrd 2513 . . . . . 6  |-  ( ph  ->  ( V  -  D
)  e.  NN )
595, 58nnaddcld 10360 . . . . 5  |-  ( ph  ->  ( A  +  ( V  -  D ) )  e.  NN )
60 nnm1nn0 10613 . . . . 5  |-  ( ( A  +  ( V  -  D ) )  e.  NN  ->  (
( A  +  ( V  -  D ) )  -  1 )  e.  NN0 )
6159, 60syl 16 . . . 4  |-  ( ph  ->  ( ( A  +  ( V  -  D
) )  -  1 )  e.  NN0 )
624, 61nn0mulcld 10633 . . 3  |-  ( ph  ->  ( W  x.  (
( A  +  ( V  -  D ) )  -  1 ) )  e.  NN0 )
63 nnnn0addcl 10602 . . 3  |-  ( ( B  e.  NN  /\  ( W  x.  (
( A  +  ( V  -  D ) )  -  1 ) )  e.  NN0 )  ->  ( B  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )  e.  NN )
642, 62, 63syl2anc 661 . 2  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )  e.  NN )
651, 64syl5eqel 2522 1  |-  ( ph  ->  T  e.  NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2710    u. cun 3321    C_ wss 3323   ifcif 3786   {csn 3872   class class class wbr 4287    e. cmpt 4345   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   -->wf 5409   ` cfv 5413  (class class class)co 6086    ^m cmap 7206   Fincfn 7302   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    - cmin 9587   NNcn 10314   2c2 10363   NN0cn0 10571   ZZ>=cuz 10853   ...cfz 11429   #chash 12095  APcvdwa 14018
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-vdwap 14021
This theorem is referenced by:  vdwlem6  14039
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