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

Theorem ser1const 11862
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 5691 . . . . 5  |-  ( j  =  1  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 1 ) )
2 oveq1 6098 . . . . 5  |-  ( j  =  1  ->  (
j  x.  A )  =  ( 1  x.  A ) )
31, 2eqeq12d 2457 . . . 4  |-  ( j  =  1  ->  (
(  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 j )  =  ( j  x.  A
)  <->  (  seq 1
(  +  ,  ( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) ) )
43imbi2d 316 . . 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 5691 . . . . 5  |-  ( j  =  k  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 k ) )
6 oveq1 6098 . . . . 5  |-  ( j  =  k  ->  (
j  x.  A )  =  ( k  x.  A ) )
75, 6eqeq12d 2457 . . . 4  |-  ( j  =  k  ->  (
(  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 j )  =  ( j  x.  A
)  <->  (  seq 1
(  +  ,  ( NN  X.  { A } ) ) `  k )  =  ( k  x.  A ) ) )
87imbi2d 316 . . 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 5691 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
10 oveq1 6098 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  A )  =  ( ( k  +  1 )  x.  A ) )
119, 10eqeq12d 2457 . . . 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 316 . . 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 5691 . . . . 5  |-  ( j  =  N  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 N ) )
14 oveq1 6098 . . . . 5  |-  ( j  =  N  ->  (
j  x.  A )  =  ( N  x.  A ) )
1513, 14eqeq12d 2457 . . . 4  |-  ( j  =  N  ->  (
(  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 j )  =  ( j  x.  A
)  <->  (  seq 1
(  +  ,  ( NN  X.  { A } ) ) `  N )  =  ( N  x.  A ) ) )
1615imbi2d 316 . . 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 10676 . . . 4  |-  1  e.  ZZ
18 1nn 10333 . . . . . 6  |-  1  e.  NN
19 fvconst2g 5931 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  NN )  ->  ( ( NN  X.  { A } ) ` 
1 )  =  A )
2018, 19mpan2 671 . . . . 5  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  A )
21 mulid2 9384 . . . . 5  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2220, 21eqtr4d 2478 . . . 4  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  ( 1  x.  A ) )
2317, 22seq1i 11820 . . 3  |-  ( A  e.  CC  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) )
24 oveq1 6098 . . . . . 6  |-  ( (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  =  ( k  x.  A
)  ->  ( (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  k )  +  A
)  =  ( ( k  x.  A )  +  A ) )
25 seqp1 11821 . . . . . . . . . 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 10896 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2725, 26eleq2s 2535 . . . . . . . . 9  |-  ( k  e.  NN  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  ( k  +  1 ) )  =  ( (  seq 1 (  +  ,  ( NN 
X.  { A }
) ) `  k
)  +  ( ( NN  X.  { A } ) `  (
k  +  1 ) ) ) )
2827adantl 466 . . . . . . . 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 10334 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
30 fvconst2g 5931 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( k  +  1 )  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( k  +  1 ) )  =  A )
3129, 30sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( k  +  1 ) )  =  A )
3231oveq2d 6107 . . . . . . . 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 2475 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  +  A ) )
34 nncn 10330 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  CC )
35 id 22 . . . . . . . . 9  |-  ( A  e.  CC  ->  A  e.  CC )
36 ax-1cn 9340 . . . . . . . . . 10  |-  1  e.  CC
37 adddir 9377 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  CC )  ->  (
( k  +  1 )  x.  A )  =  ( ( k  x.  A )  +  ( 1  x.  A
) ) )
3836, 37mp3an2 1302 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  A  e.  CC )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  ( 1  x.  A ) ) )
3934, 35, 38syl2anr 478 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  ( 1  x.  A ) ) )
4021adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( 1  x.  A
)  =  A )
4140oveq2d 6107 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  x.  A )  +  ( 1  x.  A ) )  =  ( ( k  x.  A )  +  A ) )
4239, 41eqtrd 2475 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  A ) )
4333, 42eqeq12d 2457 . . . . . 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 221 . . . . 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 435 . . . 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 26 . . 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 10340 . 2  |-  ( N  e.  NN  ->  ( A  e.  CC  ->  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 N )  =  ( N  x.  A
) ) )
4847impcom 430 1  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  +  ,  ( NN 
X.  { A }
) ) `  N
)  =  ( N  x.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3877    X. cxp 4838   ` cfv 5418  (class class class)co 6091   CCcc 9280   1c1 9283    + caddc 9285    x. cmul 9287   NNcn 10322   ZZ>=cuz 10861    seqcseq 11806
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 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-seq 11807
This theorem is referenced by:  fsumconst  13257  vitalilem4  21091  ovoliunnfl  28433  voliunnfl  28435
  Copyright terms: Public domain W3C validator