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Theorem vdwapfval 14696
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 997 . . . . . . 7  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  k  =  K )
21oveq1d 6292 . . . . . 6  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  (
k  -  1 )  =  ( K  - 
1 ) )
32oveq2d 6293 . . . . 5  |-  ( ( k  =  K  /\  a  e.  NN  /\  d  e.  NN )  ->  (
0 ... ( k  - 
1 ) )  =  ( 0 ... ( K  -  1 ) ) )
43mpteq1d 4475 . . . 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 5050 . . 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 6341 . 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 14693 . 2  |- AP  =  ( k  e.  NN0  |->  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( k  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) )
8 nnex 10581 . . 3  |-  NN  e.  _V
98, 8mpt2ex 6860 . 2  |-  ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) )  e.  _V
106, 7, 9fvmpt 5931 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 974    = wceq 1405    e. wcel 1842    |-> cmpt 4452   ran crn 4823   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    - cmin 9840   NNcn 10575   NN0cn0 10835   ...cfz 11724  APcvdwa 14690
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 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-i2m1 9589  ax-1ne0 9590  ax-rrecex 9593  ax-cnre 9594
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-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-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-nn 10576  df-vdwap 14693
This theorem is referenced by:  vdwapf  14697  vdwapval  14698
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