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

Theorem vdwapf 13021
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 10824 . . . . . . . . . 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 9974 . . . . . . . . . 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 10025 . . . . . . . 8  |-  ( ( ( a  e.  NN  /\  d  e.  NN )  /\  m  e.  ( 0 ... ( K  -  1 ) ) )  ->  ( m  x.  d )  e.  NN0 )
7 nnnn0addcl 9997 . . . . . . . 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 2285 . . . . . . 7  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )
108, 9fmptd 5686 . . . . . 6  |-  ( ( a  e.  NN  /\  d  e.  NN )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) : ( 0 ... ( K  -  1 ) ) --> NN )
11 frn 5397 . . . . . 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 9754 . . . . . 6  |-  NN  e.  _V
1413elpw2 4177 . . . . 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 2611 . . 3  |-  A. a  e.  NN  A. d  e.  NN  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) )  e.  ~P NN
17 eqid 2285 . . . 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 6193 . . 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 13020 . . 3  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
2120feq1d 5381 . 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 1686   A.wral 2545    C_ wss 3154   ~Pcpw 3627    e. cmpt 4079    X. cxp 4689   ran crn 4692   -->wf 5253   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    - cmin 9039   NNcn 9748   NN0cn0 9967   ...cfz 10784  APcvdwa 13014
This theorem is referenced by:  vdwmc  13027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-vdwap 13017
  Copyright terms: Public domain W3C validator