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Theorem ser1const 11371
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 5720 . . . . 5  |-  ( j  =  1  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 1 ) )
2 oveq1 6080 . . . . 5  |-  ( j  =  1  ->  (
j  x.  A )  =  ( 1  x.  A ) )
31, 2eqeq12d 2449 . . . 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 5720 . . . . 5  |-  ( j  =  k  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k ) )
6 oveq1 6080 . . . . 5  |-  ( j  =  k  ->  (
j  x.  A )  =  ( k  x.  A ) )
75, 6eqeq12d 2449 . . . 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 5720 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
10 oveq1 6080 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  A )  =  ( ( k  +  1 )  x.  A ) )
119, 10eqeq12d 2449 . . . 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 5720 . . . . 5  |-  ( j  =  N  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 N ) )
14 oveq1 6080 . . . . 5  |-  ( j  =  N  ->  (
j  x.  A )  =  ( N  x.  A ) )
1513, 14eqeq12d 2449 . . . 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 10303 . . . 4  |-  1  e.  ZZ
18 1nn 10003 . . . . . 6  |-  1  e.  NN
19 fvconst2g 5937 . . . . . 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 9081 . . . . 5  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2220, 21eqtr4d 2470 . . . 4  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  ( 1  x.  A ) )
2317, 22seq1i 11329 . . 3  |-  ( A  e.  CC  ->  (  seq  1 (  +  , 
( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) )
24 oveq1 6080 . . . . . 6  |-  ( (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  =  ( k  x.  A
)  ->  ( (  seq  1 (  +  , 
( NN  X.  { A } ) ) `  k )  +  A
)  =  ( ( k  x.  A )  +  A ) )
25 seqp1 11330 . . . . . . . . . 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 10513 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2725, 26eleq2s 2527 . . . . . . . . 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 10004 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
30 fvconst2g 5937 . . . . . . . . . 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 6089 . . . . . . . 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 2467 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  (  seq  1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( (  seq  1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  +  A ) )
34 nncn 10000 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  CC )
35 id 20 . . . . . . . . 9  |-  ( A  e.  CC  ->  A  e.  CC )
36 ax-1cn 9040 . . . . . . . . . 10  |-  1  e.  CC
37 adddir 9075 . . . . . . . . . 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 6089 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  x.  A )  +  ( 1  x.  A ) )  =  ( ( k  x.  A )  +  A ) )
4239, 41eqtrd 2467 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  A ) )
4333, 42eqeq12d 2449 . . . . . 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 10010 . 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 1652    e. wcel 1725   {csn 3806    X. cxp 4868   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983    + caddc 8985    x. cmul 8987   NNcn 9992   ZZ>=cuz 10480    seq cseq 11315
This theorem is referenced by:  fsumconst  12565  vitalilem4  19495  ovoliunnfl  26238  voliunnfl  26240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316
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