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Theorem vdwapf 13371
Description: The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
Assertion
Ref Expression
vdwapf  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )

Proof of Theorem vdwapf
Dummy variables  a 
d  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 732 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  a  e.  NN )
2 elfznn0 11114 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  m  e.  NN0 )
32adantl 454 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  m  e.  NN0 )
4 nnnn0 10259 . . . . . . . . . 10  |-  ( d  e.  NN  ->  d  e.  NN0 )
54ad2antlr 709 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  d  e.  NN0 )
63, 5nn0mulcld 10310 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( m  x.  d )  e.  NN0 )
7 nnnn0addcl 10282 . . . . . . . 8  |-  ( ( a  e.  NN  /\  ( m  x.  d
)  e.  NN0 )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
81, 6, 7syl2anc 644 . . . . . . 7  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
9 eqid 2442 . . . . . . 7  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )
108, 9fmptd 5922 . . . . . 6  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) : ( 0 ... ( K  -  1 ) ) --> NN )
11 frn 5626 . . . . . 6  |-  ( ( m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) : ( 0 ... ( K  - 
1 ) ) --> NN 
->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  C_  NN )
1210, 11syl 16 . . . . 5  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  C_  NN )
13 nnex 10037 . . . . . 6  |-  NN  e.  _V
1413elpw2 4393 . . . . 5  |-  ( ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) )  e.  ~P NN 
<->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  C_  NN )
1512, 14sylibr 205 . . . 4  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  e. 
~P NN )
1615rgen2a 2778 . . 3  |-  A. a  e.  NN  A. d  e.  NN  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  e.  ~P NN
17 eqid 2442 . . . 4  |-  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) )
1817fmpt2 6447 . . 3  |-  ( A. a  e.  NN  A. d  e.  NN  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  e.  ~P NN  <->  ( a  e.  NN ,  d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) ) ) : ( NN  X.  NN ) --> ~P NN )
1916, 18mpbi 201 . 2  |-  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) : ( NN  X.  NN ) --> ~P NN
20 vdwapfval 13370 . . 3  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
2120feq1d 5609 . 2  |-  ( K  e.  NN0  ->  ( (AP
`  K ) : ( NN  X.  NN )
--> ~P NN  <->  ( a  e.  NN ,  d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) ) ) : ( NN  X.  NN ) --> ~P NN ) )
2219, 21mpbiri 226 1  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727   A.wral 2711    C_ wss 3306   ~Pcpw 3823    e. cmpt 4291    X. cxp 4905   ran crn 4908   -->wf 5479   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112   0cc0 9021   1c1 9022    + caddc 9024    x. cmul 9026    - cmin 9322   NNcn 10031   NN0cn0 10252   ...cfz 11074  APcvdwa 13364
This theorem is referenced by:  vdwmc  13377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-vdwap 13367
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