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Theorem vdwapval 14346
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 14344 . . . . . . 7  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
213ad2ant1 1017 . . . . . 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 6299 . . . . 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 6290 . . . . . . . . . 10  |-  ( d  =  D  ->  (
m  x.  d )  =  ( m  x.  D ) )
5 oveq12 6291 . . . . . . . . . 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 4534 . . . . . . . 8  |-  ( ( a  =  A  /\  d  =  D )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) ) )
87rneqd 5228 . . . . . . 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 2467 . . . . . . 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 6307 . . . . . . . . 9  |-  ( 0 ... ( K  - 
1 ) )  e. 
_V
1110mptex 6129 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
1211rnex 6715 . . . . . . 7  |-  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
138, 9, 12ovmpt2a 6415 . . . . . 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 1014 . . . . 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 2508 . . . 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 2467 . . . . 5  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D
) ) )
1716rnmpt 5246 . . . 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 2524 . . 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 2537 . 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 22 . . . . 5  |-  ( X  =  ( A  +  ( m  x.  D
) )  ->  X  =  ( A  +  ( m  x.  D
) ) )
21 ovex 6307 . . . . 5  |-  ( A  +  ( m  x.  D ) )  e. 
_V
2220, 21syl6eqel 2563 . . . 4  |-  ( X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
2322rexlimivw 2952 . . 3  |-  ( E. m  e.  ( 0 ... ( K  - 
1 ) ) X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
24 eqeq1 2471 . . . 4  |-  ( x  =  X  ->  (
x  =  ( A  +  ( m  x.  D ) )  <->  X  =  ( A  +  (
m  x.  D ) ) ) )
2524rexbidv 2973 . . 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 3257 . 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 261 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113    |-> cmpt 4505   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   NNcn 10532   NN0cn0 10791   ...cfz 11668  APcvdwa 14338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-nn 10533  df-vdwap 14341
This theorem is referenced by:  vdwapun  14347  vdwap0  14349  vdwmc2  14352  vdwlem1  14354  vdwlem2  14355  vdwlem6  14359  vdwlem8  14361
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