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Theorem vdwapval 14145
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 14143 . . . . . . 7  |-  ( K  e.  NN0  ->  (AP `  K )  =  ( a  e.  NN , 
d  e.  NN  |->  ran  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) ) ) )
213ad2ant1 1009 . . . . . 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 6210 . . . . 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 6201 . . . . . . . . . 10  |-  ( d  =  D  ->  (
m  x.  d )  =  ( m  x.  D ) )
5 oveq12 6202 . . . . . . . . . 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 4480 . . . . . . . 8  |-  ( ( a  =  A  /\  d  =  D )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) ) )
87rneqd 5168 . . . . . . 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 2451 . . . . . . 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 6218 . . . . . . . . 9  |-  ( 0 ... ( K  - 
1 ) )  e. 
_V
1110mptex 6050 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
1211rnex 6615 . . . . . . 7  |-  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
138, 9, 12ovmpt2a 6324 . . . . . 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 1006 . . . . 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 2492 . . . 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 2451 . . . . 5  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D
) ) )
1716rnmpt 5186 . . . 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 2508 . . 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 2521 . 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 6218 . . . . 5  |-  ( A  +  ( m  x.  D ) )  e. 
_V
2220, 21syl6eqel 2547 . . . 4  |-  ( X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
2322rexlimivw 2936 . . 3  |-  ( E. m  e.  ( 0 ... ( K  - 
1 ) ) X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
24 eqeq1 2455 . . . 4  |-  ( x  =  X  ->  (
x  =  ( A  +  ( m  x.  D ) )  <->  X  =  ( A  +  (
m  x.  D ) ) ) )
2524rexbidv 2855 . . 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 3213 . 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 965    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796   _Vcvv 3071    |-> cmpt 4451   ran crn 4942   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391    - cmin 9699   NNcn 10426   NN0cn0 10683   ...cfz 11547  APcvdwa 14137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-i2m1 9454  ax-1ne0 9455  ax-rrecex 9458  ax-cnre 9459
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-nn 10427  df-vdwap 14140
This theorem is referenced by:  vdwapun  14146  vdwap0  14148  vdwmc2  14151  vdwlem1  14153  vdwlem2  14154  vdwlem6  14158  vdwlem8  14160
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