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Theorem vdwapval 14026
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 14024 . . . . . . 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 6103 . . . . 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 6094 . . . . . . . . . 10  |-  ( d  =  D  ->  (
m  x.  d )  =  ( m  x.  D ) )
5 oveq12 6095 . . . . . . . . . 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 4374 . . . . . . . 8  |-  ( ( a  =  A  /\  d  =  D )  ->  ( m  e.  ( 0 ... ( K  -  1 ) ) 
|->  ( a  +  ( m  x.  d ) ) )  =  ( m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) ) )
87rneqd 5062 . . . . . . 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 2438 . . . . . . 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 6111 . . . . . . . . 9  |-  ( 0 ... ( K  - 
1 ) )  e. 
_V
1110mptex 5943 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
1211rnex 6507 . . . . . . 7  |-  ran  (
m  e.  ( 0 ... ( K  - 
1 ) )  |->  ( A  +  ( m  x.  D ) ) )  e.  _V
138, 9, 12ovmpt2a 6216 . . . . . 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 2470 . . . 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 2438 . . . . 5  |-  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D ) ) )  =  ( m  e.  ( 0 ... ( K  -  1 ) )  |->  ( A  +  ( m  x.  D
) ) )
1716rnmpt 5080 . . . 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 2486 . . 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 2505 . 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 6111 . . . . 5  |-  ( A  +  ( m  x.  D ) )  e. 
_V
2220, 21syl6eqel 2526 . . . 4  |-  ( X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
2322rexlimivw 2832 . . 3  |-  ( E. m  e.  ( 0 ... ( K  - 
1 ) ) X  =  ( A  +  ( m  x.  D
) )  ->  X  e.  _V )
24 eqeq1 2444 . . . 4  |-  ( x  =  X  ->  (
x  =  ( A  +  ( m  x.  D ) )  <->  X  =  ( A  +  (
m  x.  D ) ) ) )
2524rexbidv 2731 . . 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 3108 . 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 1369    e. wcel 1756   {cab 2424   E.wrex 2711   _Vcvv 2967    e. cmpt 4345   ran crn 4836   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587   NNcn 10314   NN0cn0 10571   ...cfz 11429  APcvdwa 14018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-i2m1 9342  ax-1ne0 9343  ax-rrecex 9346  ax-cnre 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-nn 10315  df-vdwap 14021
This theorem is referenced by:  vdwapun  14027  vdwap0  14029  vdwmc2  14032  vdwlem1  14034  vdwlem2  14035  vdwlem6  14039  vdwlem8  14041
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