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Theorem vdwapfval 14337
Description: Define the arithmetic progression function, which takes as input a length  k, a start point  a, and a step  d and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
Assertion
Ref Expression
vdwapfval  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
Distinct variable group:    a, d, m, K

Proof of Theorem vdwapfval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 simp1 991 . . . . . . 7  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  k  =  K )
21oveq1d 6290 . . . . . 6  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  (
k  -  1 )  =  ( K  - 
1 ) )
32oveq2d 6291 . . . . 5  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  (
0 ... ( k  - 
1 ) )  =  ( 0 ... ( K  -  1 ) ) )
43mpteq1d 4521 . . . 4  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  (
m  e.  ( 0 ... ( k  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d ) ) ) )
54rneqd 5221 . . 3  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  ran  ( m  e.  (
0 ... ( k  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) )  =  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) )
65mpt2eq3dva 6336 . 2  |-  ( k  =  K  ->  (
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 ) ) ) ) )
7 df-vdwap 14334 . 2  |- AP  =  ( k  e.  NN0  |->  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( k  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) )
8 nnex 10531 . . 3  |-  NN  e.  _V
98, 8mpt2ex 6850 . 2  |-  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) )  e.  _V
106, 7, 9fvmpt 5941 1  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762    |-> cmpt 4498   ran crn 4993   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9794   NNcn 10525   NN0cn0 10784   ...cfz 11661  APcvdwa 14331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-nn 10526  df-vdwap 14334
This theorem is referenced by:  vdwapf  14338  vdwapval  14339
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