MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ser1const Unicode version

Theorem ser1const 11102
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 5525 . . . . 5  |-  ( j  =  1  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 1 ) )
2 oveq1 5865 . . . . 5  |-  ( j  =  1  ->  (
j  x.  A )  =  ( 1  x.  A ) )
31, 2eqeq12d 2297 . . . 4  |-  ( j  =  1  ->  (
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A )  <->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) ) )
43imbi2d 307 . . 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 5525 . . . . 5  |-  ( j  =  k  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k ) )
6 oveq1 5865 . . . . 5  |-  ( j  =  k  ->  (
j  x.  A )  =  ( k  x.  A ) )
75, 6eqeq12d 2297 . . . 4  |-  ( j  =  k  ->  (
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A )  <->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  k )  =  ( k  x.  A ) ) )
87imbi2d 307 . . 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 5525 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
10 oveq1 5865 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  A )  =  ( ( k  +  1 )  x.  A ) )
119, 10eqeq12d 2297 . . . 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 307 . . 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 5525 . . . . 5  |-  ( j  =  N  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 N ) )
14 oveq1 5865 . . . . 5  |-  ( j  =  N  ->  (
j  x.  A )  =  ( N  x.  A ) )
1513, 14eqeq12d 2297 . . . 4  |-  ( j  =  N  ->  (
(  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  j
)  =  ( j  x.  A )  <->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  N )  =  ( N  x.  A ) ) )
1615imbi2d 307 . . 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 10053 . . . 4  |-  1  e.  ZZ
18 1nn 9757 . . . . . 6  |-  1  e.  NN
19 fvconst2g 5727 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2018, 19mpan2 652 . . . . 5  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  A )
21 mulid2 8836 . . . . 5  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2220, 21eqtr4d 2318 . . . 4  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  ( 1  x.  A ) )
2317, 22seq1i 11060 . . 3  |-  ( A  e.  CC  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) )
24 oveq1 5865 . . . . . 6  |-  ( (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  =  ( k  x.  A
)  ->  ( (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  k )  +  A
)  =  ( ( k  x.  A )  +  A ) )
25 seqp1 11061 . . . . . . . . . 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 10263 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2725, 26eleq2s 2375 . . . . . . . . 9  |-  ( k  e.  NN  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  +  ( ( NN  X.  { A } ) `  (
k  +  1 ) ) ) )
2827adantl 452 . . . . . . . 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 9758 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
30 fvconst2g 5727 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
3129, 30sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( k  +  1 ) )  =  A )
3231oveq2d 5874 . . . . . . . 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 2315 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  +  A ) )
34 nncn 9754 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  CC )
35 id 19 . . . . . . . . 9  |-  ( A  e.  CC  ->  A  e.  CC )
36 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
37 adddir 8830 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  CC )  ->  (
( k  +  1 )  x.  A )  =  ( ( k  x.  A )  +  ( 1  x.  A
) ) )
3836, 37mp3an2 1265 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  A  e.  CC )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  ( 1  x.  A ) ) )
3934, 35, 38syl2anr 464 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  ( 1  x.  A ) ) )
4021adantr 451 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( 1  x.  A
)  =  A )
4140oveq2d 5874 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  x.  A )  +  ( 1  x.  A ) )  =  ( ( k  x.  A )  +  A ) )
4239, 41eqtrd 2315 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  A ) )
4333, 42eqeq12d 2297 . . . . . 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 212 . . . . 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 424 . . . 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 23 . . 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 9764 . 2  |-  ( N  e.  NN  ->  ( A  e.  CC  ->  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 N )  =  ( N  x.  A
) ) )
4847impcom 419 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 358    = wceq 1623    e. wcel 1684   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   ZZ>=cuz 10230    seq cseq 11046
This theorem is referenced by:  fsumconst  12252  vitalilem4  18966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047
  Copyright terms: Public domain W3C validator