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Theorem ser1const 11306
Description: Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
Assertion
Ref Expression
ser1const  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  N
)  =  ( N  x.  A ) )

Proof of Theorem ser1const
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5668 . . . . 5  |-  ( j  =  1  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 1 ) )
2 oveq1 6027 . . . . 5  |-  ( j  =  1  ->  (
j  x.  A )  =  ( 1  x.  A ) )
31, 2eqeq12d 2401 . . . 4  |-  ( j  =  1  ->  (
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A )  <->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) ) )
43imbi2d 308 . . 3  |-  ( j  =  1  ->  (
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A ) )  <-> 
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  1
)  =  ( 1  x.  A ) ) ) )
5 fveq2 5668 . . . . 5  |-  ( j  =  k  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k ) )
6 oveq1 6027 . . . . 5  |-  ( j  =  k  ->  (
j  x.  A )  =  ( k  x.  A ) )
75, 6eqeq12d 2401 . . . 4  |-  ( j  =  k  ->  (
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A )  <->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  k )  =  ( k  x.  A ) ) )
87imbi2d 308 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A ) )  <-> 
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  =  ( k  x.  A ) ) ) )
9 fveq2 5668 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
10 oveq1 6027 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  A )  =  ( ( k  +  1 )  x.  A ) )
119, 10eqeq12d 2401 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A )  <->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( ( k  +  1 )  x.  A ) ) )
1211imbi2d 308 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A ) )  <-> 
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( ( k  +  1 )  x.  A ) ) ) )
13 fveq2 5668 . . . . 5  |-  ( j  =  N  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 N ) )
14 oveq1 6027 . . . . 5  |-  ( j  =  N  ->  (
j  x.  A )  =  ( N  x.  A ) )
1513, 14eqeq12d 2401 . . . 4  |-  ( j  =  N  ->  (
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A )  <->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  N )  =  ( N  x.  A ) ) )
1615imbi2d 308 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A ) )  <-> 
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  N
)  =  ( N  x.  A ) ) ) )
17 1z 10243 . . . 4  |-  1  e.  ZZ
18 1nn 9943 . . . . . 6  |-  1  e.  NN
19 fvconst2g 5884 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2018, 19mpan2 653 . . . . 5  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  A )
21 mulid2 9022 . . . . 5  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2220, 21eqtr4d 2422 . . . 4  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  ( 1  x.  A ) )
2317, 22seq1i 11264 . . 3  |-  ( A  e.  CC  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) )
24 oveq1 6027 . . . . . 6  |-  ( (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  =  ( k  x.  A
)  ->  ( (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  k )  +  A
)  =  ( ( k  x.  A )  +  A ) )
25 seqp1 11265 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  1
)  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  +  ( ( NN  X.  { A } ) `  (
k  +  1 ) ) ) )
26 nnuz 10453 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2725, 26eleq2s 2479 . . . . . . . . 9  |-  ( k  e.  NN  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  +  ( ( NN  X.  { A } ) `  (
k  +  1 ) ) ) )
2827adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  +  ( ( NN  X.  { A } ) `  ( k  +  1 ) ) ) )
29 peano2nn 9944 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
30 fvconst2g 5884 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
3129, 30sylan2 461 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( k  +  1 ) )  =  A )
3231oveq2d 6036 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( (  seq  1
(  +  ,  ( NN  X.  { A } ) ) `  k )  +  ( ( NN  X.  { A } ) `  (
k  +  1 ) ) )  =  ( (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  +  A ) )
3328, 32eqtrd 2419 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  +  A ) )
34 nncn 9940 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  CC )
35 id 20 . . . . . . . . 9  |-  ( A  e.  CC  ->  A  e.  CC )
36 ax-1cn 8981 . . . . . . . . . 10  |-  1  e.  CC
37 adddir 9016 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  CC )  ->  (
( k  +  1 )  x.  A )  =  ( ( k  x.  A )  +  ( 1  x.  A
) ) )
3836, 37mp3an2 1267 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  A  e.  CC )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  ( 1  x.  A ) ) )
3934, 35, 38syl2anr 465 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  ( 1  x.  A ) ) )
4021adantr 452 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( 1  x.  A
)  =  A )
4140oveq2d 6036 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  x.  A )  +  ( 1  x.  A ) )  =  ( ( k  x.  A )  +  A ) )
4239, 41eqtrd 2419 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  A ) )
4333, 42eqeq12d 2401 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( (  seq  1
(  +  ,  ( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( ( k  +  1 )  x.  A )  <-> 
( (  seq  1
(  +  ,  ( NN  X.  { A } ) ) `  k )  +  A
)  =  ( ( k  x.  A )  +  A ) ) )
4424, 43syl5ibr 213 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( (  seq  1
(  +  ,  ( NN  X.  { A } ) ) `  k )  =  ( k  x.  A )  ->  (  seq  1
(  +  ,  ( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( ( k  +  1 )  x.  A ) ) )
4544expcom 425 . . . 4  |-  ( k  e.  NN  ->  ( A  e.  CC  ->  ( (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  =  ( k  x.  A )  -> 
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( ( k  +  1 )  x.  A ) ) ) )
4645a2d 24 . . 3  |-  ( k  e.  NN  ->  (
( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  =  ( k  x.  A ) )  ->  ( A  e.  CC  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( ( k  +  1 )  x.  A ) ) ) )
474, 8, 12, 16, 23, 46nnind 9950 . 2  |-  ( N  e.  NN  ->  ( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 N )  =  ( N  x.  A
) ) )
4847impcom 420 1  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  N
)  =  ( N  x.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3757    X. cxp 4816   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924    + caddc 8926    x. cmul 8928   NNcn 9932   ZZ>=cuz 10420    seq cseq 11250
This theorem is referenced by:  fsumconst  12500  vitalilem4  19370  ovoliunnfl  25953  voliunnfl  25955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-seq 11251
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