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Theorem vdwapf 13303
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 731 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  a  e.  NN )
2 elfznn0 11047 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  m  e.  NN0 )
32adantl 453 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  m  e.  NN0 )
4 nnnn0 10192 . . . . . . . . . 10  |-  ( d  e.  NN  ->  d  e.  NN0 )
54ad2antlr 708 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  d  e.  NN0 )
63, 5nn0mulcld 10243 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( m  x.  d )  e.  NN0 )
7 nnnn0addcl 10215 . . . . . . . 8  |-  ( ( a  e.  NN  /\  ( m  x.  d
)  e.  NN0 )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
81, 6, 7syl2anc 643 . . . . . . 7  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
9 eqid 2412 . . . . . . 7  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )
108, 9fmptd 5860 . . . . . 6  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) : ( 0 ... ( K  -  1 ) ) --> NN )
11 frn 5564 . . . . . 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 9970 . . . . . 6  |-  NN  e.  _V
1413elpw2 4332 . . . . 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 204 . . . 4  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  e. 
~P NN )
1615rgen2a 2740 . . 3  |-  A. a  e.  NN  A. d  e.  NN  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  e.  ~P NN
17 eqid 2412 . . . 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 6385 . . 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 200 . 2  |-  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) : ( NN  X.  NN ) --> ~P NN
20 vdwapfval 13302 . . 3  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
2120feq1d 5547 . 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 225 1  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   A.wral 2674    C_ wss 3288   ~Pcpw 3767    e. cmpt 4234    X. cxp 4843   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959    - cmin 9255   NNcn 9964   NN0cn0 10185   ...cfz 11007  APcvdwa 13296
This theorem is referenced by:  vdwmc  13309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-vdwap 13299
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