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Theorem ser1const 12143
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 5872 . . . . 5  |-  ( j  =  1  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 1 ) )
2 oveq1 6302 . . . . 5  |-  ( j  =  1  ->  (
j  x.  A )  =  ( 1  x.  A ) )
31, 2eqeq12d 2489 . . . 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 5872 . . . . 5  |-  ( j  =  k  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 k ) )
6 oveq1 6302 . . . . 5  |-  ( j  =  k  ->  (
j  x.  A )  =  ( k  x.  A ) )
75, 6eqeq12d 2489 . . . 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 5872 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 ( k  +  1 ) ) )
10 oveq1 6302 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  A )  =  ( ( k  +  1 )  x.  A ) )
119, 10eqeq12d 2489 . . . 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 5872 . . . . 5  |-  ( j  =  N  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  j )  =  (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 N ) )
14 oveq1 6302 . . . . 5  |-  ( j  =  N  ->  (
j  x.  A )  =  ( N  x.  A ) )
1513, 14eqeq12d 2489 . . . 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 10906 . . . 4  |-  1  e.  ZZ
18 1nn 10559 . . . . . 6  |-  1  e.  NN
19 fvconst2g 6125 . . . . . 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 9606 . . . . 5  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2220, 21eqtr4d 2511 . . . 4  |-  ( A  e.  CC  ->  (
( NN  X.  { A } ) `  1
)  =  ( 1  x.  A ) )
2317, 22seq1i 12101 . . 3  |-  ( A  e.  CC  ->  (  seq 1 (  +  , 
( NN  X.  { A } ) ) ` 
1 )  =  ( 1  x.  A ) )
24 oveq1 6302 . . . . . 6  |-  ( (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  =  ( k  x.  A
)  ->  ( (  seq 1 (  +  , 
( NN  X.  { A } ) ) `  k )  +  A
)  =  ( ( k  x.  A )  +  A ) )
25 seqp1 12102 . . . . . . . . . 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 11129 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2725, 26eleq2s 2575 . . . . . . . . 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 10560 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
30 fvconst2g 6125 . . . . . . . . . 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 6311 . . . . . . . 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 2508 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  ( NN 
X.  { A }
) ) `  (
k  +  1 ) )  =  ( (  seq 1 (  +  ,  ( NN  X.  { A } ) ) `
 k )  +  A ) )
34 nncn 10556 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  CC )
35 id 22 . . . . . . . . 9  |-  ( A  e.  CC  ->  A  e.  CC )
36 ax-1cn 9562 . . . . . . . . . 10  |-  1  e.  CC
37 adddir 9599 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  CC )  ->  (
( k  +  1 )  x.  A )  =  ( ( k  x.  A )  +  ( 1  x.  A
) ) )
3836, 37mp3an2 1312 . . . . . . . . 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 6311 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  x.  A )  +  ( 1  x.  A ) )  =  ( ( k  x.  A )  +  A ) )
4239, 41eqtrd 2508 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  A
)  =  ( ( k  x.  A )  +  A ) )
4333, 42eqeq12d 2489 . . . . . 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 10566 . 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 1379    e. wcel 1767   {csn 4033    X. cxp 5003   ` cfv 5594  (class class class)co 6295   CCcc 9502   1c1 9505    + caddc 9507    x. cmul 9509   NNcn 10548   ZZ>=cuz 11094    seqcseq 12087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-seq 12088
This theorem is referenced by:  fsumconst  13585  vitalilem4  21888  ovoliunnfl  29983  voliunnfl  29985
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