MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdwapf Unicode version

Theorem vdwapf 13227
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 730 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  a  e.  NN )
2 elfznn0 10975 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  ->  m  e.  NN0 )
32adantl 452 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  m  e.  NN0 )
4 nnnn0 10121 . . . . . . . . . 10  |-  ( d  e.  NN  ->  d  e.  NN0 )
54ad2antlr 707 . . . . . . . . 9  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  d  e.  NN0 )
63, 5nn0mulcld 10172 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( m  x.  d )  e.  NN0 )
7 nnnn0addcl 10144 . . . . . . . 8  |-  ( ( a  e.  NN  /\  ( m  x.  d
)  e.  NN0 )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
81, 6, 7syl2anc 642 . . . . . . 7  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( a  +  ( m  x.  d ) )  e.  NN )
9 eqid 2366 . . . . . . 7  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )
108, 9fmptd 5795 . . . . . 6  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) : ( 0 ... ( K  -  1 ) ) --> NN )
11 frn 5501 . . . . . 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 15 . . . . 5  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  C_  NN )
13 nnex 9899 . . . . . 6  |-  NN  e.  _V
1413elpw2 4277 . . . . 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 203 . . . 4  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  e. 
~P NN )
1615rgen2a 2694 . . 3  |-  A. a  e.  NN  A. d  e.  NN  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  e.  ~P NN
17 eqid 2366 . . . 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 6318 . . 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 199 . 2  |-  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) : ( NN  X.  NN ) --> ~P NN
20 vdwapfval 13226 . . 3  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
2120feq1d 5484 . 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 224 1  |-  ( K  e.  NN0  ->  (AP `  K ) : ( NN  X.  NN ) --> ~P NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715   A.wral 2628    C_ wss 3238   ~Pcpw 3714    e. cmpt 4179    X. cxp 4790   ran crn 4793   -->wf 5354   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    - cmin 9184   NNcn 9893   NN0cn0 10114   ...cfz 10935  APcvdwa 13220
This theorem is referenced by:  vdwmc  13233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-vdwap 13223
  Copyright terms: Public domain W3C validator