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Theorem vdwlem5 14029
Description: Lemma for vdw 14038. (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 10624 . . . 4  |-  ( ph  ->  W  e.  NN0 )
5 vdwlem7.a . . . . . 6  |-  ( ph  ->  A  e.  NN )
6 vdwlem3.v . . . . . . . . . 10  |-  ( ph  ->  V  e.  NN )
76nncnd 10326 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
8 vdwlem7.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  NN )
98nncnd 10326 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
107, 9subcld 9707 . . . . . . . 8  |-  ( ph  ->  ( V  -  D
)  e.  CC )
115nncnd 10326 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1210, 11npcand 9711 . . . . . . 7  |-  ( ph  ->  ( ( ( V  -  D )  -  A )  +  A
)  =  ( V  -  D ) )
137, 9, 11subsub4d 9738 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  -  D )  -  A
)  =  ( V  -  ( D  +  A ) ) )
149, 11addcomd 9559 . . . . . . . . . . 11  |-  ( ph  ->  ( D  +  A
)  =  ( A  +  D ) )
1514oveq2d 6096 . . . . . . . . . 10  |-  ( ph  ->  ( V  -  ( D  +  A )
)  =  ( V  -  ( A  +  D ) ) )
1613, 15eqtrd 2465 . . . . . . . . 9  |-  ( ph  ->  ( ( V  -  D )  -  A
)  =  ( V  -  ( A  +  D ) ) )
17 cnvimass 5177 . . . . . . . . . . . . 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 14028 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : ( 1 ... V ) --> ( R  ^m  ( 1 ... W ) ) )
22 fdm 5551 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... V ) --> ( R  ^m  ( 1 ... W ) )  ->  dom  F  =  ( 1 ... V ) )
2321, 22syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  =  ( 1 ... V ) )
2417, 23syl5sseq 3392 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F " { G } )  C_  ( 1 ... V
) )
25 vdwlem7.s . . . . . . . . . . . . 13  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' F " { G } ) )
26 ssun2 3508 . . . . . . . . . . . . . . 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 10917 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ( ZZ>= `  2
)  ->  ( K  -  1 )  e.  NN )
2927, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  -  1 )  e.  NN )
305, 8nnaddcld 10356 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  +  D
)  e.  NN )
31 vdwapid1 14019 . . . . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  +  D
)  e.  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D ) )
3326, 32sseldi 3342 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  +  D
)  e.  ( { A }  u.  (
( A  +  D
) (AP `  ( K  -  1 ) ) D ) ) )
34 eluz2b2 10915 . . . . . . . . . . . . . . . . . . . . 21  |-  ( K  e.  ( ZZ>= `  2
)  <->  ( K  e.  NN  /\  1  < 
K ) )
3534simplbi 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  NN )
3627, 35syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  K  e.  NN )
3736nncnd 10326 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  K  e.  CC )
38 ax-1cn 9328 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
39 npcan 9607 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
4037, 38, 39sylancl 655 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
4140fveq2d 5683 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (AP `  ( ( K  -  1 )  +  1 ) )  =  (AP `  K
) )
4241oveqd 6097 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( A (AP
`  K ) D ) )
43 nnm1nn0 10609 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
4436, 43syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
45 vdwapun 14018 . . . . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4742, 46eqtr3d 2467 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A (AP `  K ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4833, 47eleqtrrd 2510 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  D
)  e.  ( A (AP `  K ) D ) )
4925, 48sseldd 3345 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  D
)  e.  ( `' F " { G } ) )
5024, 49sseldd 3345 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  D
)  e.  ( 1 ... V ) )
51 elfzuz3 11437 . . . . . . . . . . 11  |-  ( ( A  +  D )  e.  ( 1 ... V )  ->  V  e.  ( ZZ>= `  ( A  +  D ) ) )
5250, 51syl 16 . . . . . . . . . 10  |-  ( ph  ->  V  e.  ( ZZ>= `  ( A  +  D
) ) )
53 uznn0sub 10880 . . . . . . . . . 10  |-  ( V  e.  ( ZZ>= `  ( A  +  D )
)  ->  ( V  -  ( A  +  D ) )  e. 
NN0 )
5452, 53syl 16 . . . . . . . . 9  |-  ( ph  ->  ( V  -  ( A  +  D )
)  e.  NN0 )
5516, 54eqeltrd 2507 . . . . . . . 8  |-  ( ph  ->  ( ( V  -  D )  -  A
)  e.  NN0 )
56 nn0nnaddcl 10599 . . . . . . . 8  |-  ( ( ( ( V  -  D )  -  A
)  e.  NN0  /\  A  e.  NN )  ->  ( ( ( V  -  D )  -  A )  +  A
)  e.  NN )
5755, 5, 56syl2anc 654 . . . . . . 7  |-  ( ph  ->  ( ( ( V  -  D )  -  A )  +  A
)  e.  NN )
5812, 57eqeltrrd 2508 . . . . . 6  |-  ( ph  ->  ( V  -  D
)  e.  NN )
595, 58nnaddcld 10356 . . . . 5  |-  ( ph  ->  ( A  +  ( V  -  D ) )  e.  NN )
60 nnm1nn0 10609 . . . . 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 10629 . . 3  |-  ( ph  ->  ( W  x.  (
( A  +  ( V  -  D ) )  -  1 ) )  e.  NN0 )
63 nnnn0addcl 10598 . . 3  |-  ( ( B  e.  NN  /\  ( W  x.  (
( A  +  ( V  -  D ) )  -  1 ) )  e.  NN0 )  ->  ( B  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )  e.  NN )
642, 62, 63syl2anc 654 . 2  |-  ( ph  ->  ( B  +  ( W  x.  ( ( A  +  ( V  -  D ) )  -  1 ) ) )  e.  NN )
651, 64syl5eqel 2517 1  |-  ( ph  ->  T  e.  NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362    e. wcel 1755   A.wral 2705    u. cun 3314    C_ wss 3316   ifcif 3779   {csn 3865   class class class wbr 4280    e. cmpt 4338   `'ccnv 4826   dom cdm 4827   ran crn 4828   "cima 4830   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   Fincfn 7298   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    - cmin 9583   NNcn 10310   2c2 10359   NN0cn0 10567   ZZ>=cuz 10849   ...cfz 11424   #chash 12087  APcvdwa 14009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-vdwap 14012
This theorem is referenced by:  vdwlem6  14030
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