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Theorem vdwlem5 14379
Description: Lemma for vdw 14388. (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 10864 . . . 4  |-  ( ph  ->  W  e.  NN0 )
5 vdwlem7.a . . . . . 6  |-  ( ph  ->  A  e.  NN )
6 vdwlem3.v . . . . . . . . . 10  |-  ( ph  ->  V  e.  NN )
76nncnd 10564 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
8 vdwlem7.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  NN )
98nncnd 10564 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
107, 9subcld 9942 . . . . . . . 8  |-  ( ph  ->  ( V  -  D
)  e.  CC )
115nncnd 10564 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1210, 11npcand 9946 . . . . . . 7  |-  ( ph  ->  ( ( ( V  -  D )  -  A )  +  A
)  =  ( V  -  D ) )
137, 9, 11subsub4d 9973 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  -  D )  -  A
)  =  ( V  -  ( D  +  A ) ) )
149, 11addcomd 9793 . . . . . . . . . . 11  |-  ( ph  ->  ( D  +  A
)  =  ( A  +  D ) )
1514oveq2d 6311 . . . . . . . . . 10  |-  ( ph  ->  ( V  -  ( D  +  A )
)  =  ( V  -  ( A  +  D ) ) )
1613, 15eqtrd 2508 . . . . . . . . 9  |-  ( ph  ->  ( ( V  -  D )  -  A
)  =  ( V  -  ( A  +  D ) ) )
17 cnvimass 5363 . . . . . . . . . . . . 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 14378 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : ( 1 ... V ) --> ( R  ^m  ( 1 ... W ) ) )
22 fdm 5741 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... V ) --> ( R  ^m  ( 1 ... W ) )  ->  dom  F  =  ( 1 ... V ) )
2321, 22syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  F  =  ( 1 ... V ) )
2417, 23syl5sseq 3557 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F " { G } )  C_  ( 1 ... V
) )
25 vdwlem7.s . . . . . . . . . . . . 13  |-  ( ph  ->  ( A (AP `  K ) D ) 
C_  ( `' F " { G } ) )
26 ssun2 3673 . . . . . . . . . . . . . . 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 11168 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ( ZZ>= `  2
)  ->  ( K  -  1 )  e.  NN )
2927, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  -  1 )  e.  NN )
305, 8nnaddcld 10594 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  +  D
)  e.  NN )
31 vdwapid1 14369 . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  +  D
)  e.  ( ( A  +  D ) (AP `  ( K  -  1 ) ) D ) )
3326, 32sseldi 3507 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  +  D
)  e.  ( { A }  u.  (
( A  +  D
) (AP `  ( K  -  1 ) ) D ) ) )
34 eluz2b2 11166 . . . . . . . . . . . . . . . . . . . . 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 10564 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  K  e.  CC )
38 ax-1cn 9562 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
39 npcan 9841 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
4037, 38, 39sylancl 662 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
4140fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (AP `  ( ( K  -  1 )  +  1 ) )  =  (AP `  K
) )
4241oveqd 6312 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( A (AP
`  K ) D ) )
43 nnm1nn0 10849 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
4436, 43syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K  -  1 )  e.  NN0 )
45 vdwapun 14368 . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A (AP `  ( ( K  - 
1 )  +  1 ) ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4742, 46eqtr3d 2510 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A (AP `  K ) D )  =  ( { A }  u.  ( ( A  +  D )
(AP `  ( K  -  1 ) ) D ) ) )
4833, 47eleqtrrd 2558 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  D
)  e.  ( A (AP `  K ) D ) )
4925, 48sseldd 3510 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  D
)  e.  ( `' F " { G } ) )
5024, 49sseldd 3510 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  D
)  e.  ( 1 ... V ) )
51 elfzuz3 11697 . . . . . . . . . . 11  |-  ( ( A  +  D )  e.  ( 1 ... V )  ->  V  e.  ( ZZ>= `  ( A  +  D ) ) )
5250, 51syl 16 . . . . . . . . . 10  |-  ( ph  ->  V  e.  ( ZZ>= `  ( A  +  D
) ) )
53 uznn0sub 11125 . . . . . . . . . 10  |-  ( V  e.  ( ZZ>= `  ( A  +  D )
)  ->  ( V  -  ( A  +  D ) )  e. 
NN0 )
5452, 53syl 16 . . . . . . . . 9  |-  ( ph  ->  ( V  -  ( A  +  D )
)  e.  NN0 )
5516, 54eqeltrd 2555 . . . . . . . 8  |-  ( ph  ->  ( ( V  -  D )  -  A
)  e.  NN0 )
56 nn0nnaddcl 10839 . . . . . . . 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 2556 . . . . . 6  |-  ( ph  ->  ( V  -  D
)  e.  NN )
595, 58nnaddcld 10594 . . . . 5  |-  ( ph  ->  ( A  +  ( V  -  D ) )  e.  NN )
60 nnm1nn0 10849 . . . . 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 10869 . . 3  |-  ( ph  ->  ( W  x.  (
( A  +  ( V  -  D ) )  -  1 ) )  e.  NN0 )
63 nnnn0addcl 10838 . . 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 2559 1  |-  ( ph  ->  T  e.  NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2817    u. cun 3479    C_ wss 3481   ifcif 3945   {csn 4033   class class class wbr 4453    |-> cmpt 4511   `'ccnv 5004   dom cdm 5005   ran crn 5006   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Fincfn 7528   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ZZ>=cuz 11094   ...cfz 11684   #chash 12385  APcvdwa 14359
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-vdwap 14362
This theorem is referenced by:  vdwlem6  14380
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