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Theorem vdwapf 14591
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 752 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  a  e.  NN )
2 elfznn0 11743 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  m  e.  NN0 )
32adantl 464 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  m  e.  NN0 )
4 nnnn0 10763 . . . . . . . . . 10  |-  ( d  e.  NN  ->  d  e.  NN0 )
54ad2antlr 725 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  d  e.  NN0 )
63, 5nn0mulcld 10818 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( m  x.  d )  e.  NN0 )
7 nnnn0addcl 10787 . . . . . . . 8  |-  ( ( a  e.  NN  /\  ( m  x.  d
)  e.  NN0 )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
81, 6, 7syl2anc 659 . . . . . . 7  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
9 eqid 2402 . . . . . . 7  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )
108, 9fmptd 5989 . . . . . 6  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) : ( 0 ... ( K  -  1 ) ) --> NN )
11 frn 5676 . . . . . 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 17 . . . . 5  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  C_  NN )
13 nnex 10502 . . . . . 6  |-  NN  e.  _V
1413elpw2 4557 . . . . 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 212 . . . 4  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  e. 
~P NN )
1615rgen2a 2830 . . 3  |-  A. a  e.  NN  A. d  e.  NN  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  e.  ~P NN
17 eqid 2402 . . . 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 6805 . . 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 208 . 2  |-  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) : ( NN  X.  NN ) --> ~P NN
20 vdwapfval 14590 . . 3  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
2120feq1d 5656 . 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 233 1  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842   A.wral 2753    C_ wss 3413   ~Pcpw 3954    |-> cmpt 4452    X. cxp 4940   ran crn 4943   -->wf 5521   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   0cc0 9442   1c1 9443    + caddc 9445    x. cmul 9447    - cmin 9761   NNcn 10496   NN0cn0 10756   ...cfz 11643  APcvdwa 14584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-vdwap 14587
This theorem is referenced by:  vdwmc  14597
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