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Theorem vdwapval 14702
Description: Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
Assertion
Ref Expression
vdwapval  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( X  e.  ( A
(AP `  K ) D )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) X  =  ( A  +  ( m  x.  D ) ) ) )
Distinct variable groups:    A, m    D, m    m, K    m, X

Proof of Theorem vdwapval
Dummy variables  a 
d  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwapfval 14700 . . . . . . 7  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
213ad2ant1 1020 . . . . . 6  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
32oveqd 6297 . . . . 5  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( A (AP `  K ) D )  =  ( A ( a  e.  NN ,  d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) ) ) D ) )
4 oveq2 6288 . . . . . . . . . 10  |-  ( d  =  D  ->  (
m  x.  d )  =  ( m  x.  D ) )
5 oveq12 6289 . . . . . . . . . 10  |-  ( ( a  =  A  /\  ( m  x.  d
)  =  ( m  x.  D ) )  ->  ( a  +  ( m  x.  d
) )  =  ( A  +  ( m  x.  D ) ) )
64, 5sylan2 474 . . . . . . . . 9  |-  ( ( a  =  A  /\  d  =  D )  ->  ( a  +  ( m  x.  d ) )  =  ( A  +  ( m  x.  D ) ) )
76mpteq2dv 4484 . . . . . . . 8  |-  ( ( a  =  A  /\  d  =  D )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) ) )
87rneqd 5053 . . . . . . 7  |-  ( ( a  =  A  /\  d  =  D )  ->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) )  =  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D
) ) ) )
9 eqid 2404 . . . . . . 7  |-  ( 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 ) ) ) )
10 ovex 6308 . . . . . . . . 9  |-  ( 0 ... ( K  - 
1 ) )  e. 
_V
1110mptex 6126 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
1211rnex 6720 . . . . . . 7  |-  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
138, 9, 12ovmpt2a 6416 . . . . . 6  |-  ( ( A  e.  NN  /\  D  e.  NN )  ->  ( A ( a  e.  NN ,  d  e.  NN  |->  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( a  +  ( m  x.  d ) ) ) ) D )  =  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) ) )
14133adant1 1017 . . . . 5  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( A ( a  e.  NN ,  d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( a  +  ( m  x.  d
) ) ) ) D )  =  ran  ( m  e.  (
0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) ) )
153, 14eqtrd 2445 . . . 4  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( A (AP `  K ) D )  =  ran  ( m  e.  (
0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) ) )
16 eqid 2404 . . . . 5  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D
) ) )
1716rnmpt 5071 . . . 4  |-  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) )  =  { x  |  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( A  +  ( m  x.  D ) ) }
1815, 17syl6eq 2461 . . 3  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( A (AP `  K ) D )  =  {
x  |  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( A  +  ( m  x.  D ) ) } )
1918eleq2d 2474 . 2  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( X  e.  ( A
(AP `  K ) D )  <->  X  e.  { x  |  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( A  +  ( m  x.  D ) ) } ) )
20 id 23 . . . . 5  |-  ( X  =  ( A  +  ( m  x.  D
) )  ->  X  =  ( A  +  ( m  x.  D
) ) )
21 ovex 6308 . . . . 5  |-  ( A  +  ( m  x.  D ) )  e. 
_V
2220, 21syl6eqel 2500 . . . 4  |-  ( X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
2322rexlimivw 2895 . . 3  |-  ( E. m  e.  ( 0 ... ( K  - 
1 ) ) X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
24 eqeq1 2408 . . . 4  |-  ( x  =  X  ->  (
x  =  ( A  +  ( m  x.  D ) )  <->  X  =  ( A  +  (
m  x.  D ) ) ) )
2524rexbidv 2920 . . 3  |-  ( x  =  X  ->  ( E. m  e.  (
0 ... ( K  - 
1 ) ) x  =  ( A  +  ( m  x.  D
) )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) X  =  ( A  +  ( m  x.  D ) ) ) )
2623, 25elab3 3205 . 2  |-  ( X  e.  { x  |  E. m  e.  ( 0 ... ( K  -  1 ) ) x  =  ( A  +  ( m  x.  D ) ) }  <->  E. m  e.  (
0 ... ( K  - 
1 ) ) X  =  ( A  +  ( m  x.  D
) ) )
2719, 26syl6bb 263 1  |-  ( ( K  e.  NN0  /\  A  e.  NN  /\  D  e.  NN )  ->  ( X  e.  ( A
(AP `  K ) D )  <->  E. m  e.  ( 0 ... ( K  -  1 ) ) X  =  ( A  +  ( m  x.  D ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   {cab 2389   E.wrex 2757   _Vcvv 3061    |-> cmpt 4455   ran crn 4826   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    - cmin 9843   NNcn 10578   NN0cn0 10838   ...cfz 11728  APcvdwa 14694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-i2m1 9592  ax-1ne0 9593  ax-rrecex 9596  ax-cnre 9597
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-nn 10579  df-vdwap 14697
This theorem is referenced by:  vdwapun  14703  vdwap0  14705  vdwmc2  14708  vdwlem1  14710  vdwlem2  14711  vdwlem6  14715  vdwlem8  14717
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