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Theorem ser1const 12857
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 6191 . . . . 5 |- ( j = 1 -> ( seq 1 ( + , ( NN X. { A } ) ) ` j ) = ( seq 1 ( + , ( NN X. { A } ) ) ` 1 ) )
2 oveq1 6657 . . . . 5 |- ( j = 1 -> ( j x. A ) = ( 1 x. A ) )
31, 2eqeq12d 2637 . . . 4 |- ( j = 1 -> ( ( seq 1 ( + , ( NN X. { A } ) ) ` j ) = ( j x. A ) <-> ( seq 1 ( + , ( NN X. { A } ) ) ` 1 ) = ( 1 x. A ) ) )
43imbi2d 330 . . 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 6191 . . . . 5 |- ( j = k -> ( seq 1 ( + , ( NN X. { A } ) ) ` j ) = ( seq 1 ( + , ( NN X. { A } ) ) ` k ) )
6 oveq1 6657 . . . . 5 |- ( j = k -> ( j x. A ) = ( k x. A ) )
75, 6eqeq12d 2637 . . . 4 |- ( j = k -> ( ( seq 1 ( + , ( NN X. { A } ) ) ` j ) = ( j x. A ) <-> ( seq 1 ( + , ( NN X. { A } ) ) ` k ) = ( k x. A ) ) )
87imbi2d 330 . . 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 6191 . . . . 5 |- ( j = ( k + 1 ) -> ( seq 1 ( + , ( NN X. { A } ) ) ` j ) = ( seq 1 ( + , ( NN X. { A } ) ) ` ( k + 1 ) ) )
10 oveq1 6657 . . . . 5 |- ( j = ( k + 1 ) -> ( j x. A ) = ( ( k + 1 ) x. A ) )
119, 10eqeq12d 2637 . . . 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 330 . . 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 6191 . . . . 5 |- ( j = N -> ( seq 1 ( + , ( NN X. { A } ) ) ` j ) = ( seq 1 ( + , ( NN X. { A } ) ) ` N ) )
14 oveq1 6657 . . . . 5 |- ( j = N -> ( j x. A ) = ( N x. A ) )
1513, 14eqeq12d 2637 . . . 4 |- ( j = N -> ( ( seq 1 ( + , ( NN X. { A } ) ) ` j ) = ( j x. A ) <-> ( seq 1 ( + , ( NN X. { A } ) ) ` N ) = ( N x. A ) ) )
1615imbi2d 330 . . 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 11407 . . . 4 |- 1 e. ZZ
18 1nn 11031 . . . . . 6 |- 1 e. NN
19 fvconst2g 6467 . . . . . 6 |- ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
2018, 19mpan2 707 . . . . 5 |- ( A e. CC -> ( ( NN X. { A } ) ` 1 ) = A )
21 mulid2 10038 . . . . 5 |- ( A e. CC -> ( 1 x. A ) = A )
2220, 21eqtr4d 2659 . . . 4 |- ( A e. CC -> ( ( NN X. { A } ) ` 1 ) = ( 1 x. A ) )
2317, 22seq1i 12815 . . 3 |- ( A e. CC -> ( seq 1 ( + , ( NN X. { A } ) ) ` 1 ) = ( 1 x. A ) )
24 oveq1 6657 . . . . . 6 |- ( ( seq 1 ( + , ( NN X. { A } ) ) ` k ) = ( k x. A ) -> ( ( seq 1 ( + , ( NN X. { A } ) ) ` k ) + A ) = ( ( k x. A ) + A ) )
25 seqp1 12816 . . . . . . . . . 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 11723 . . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
2725, 26eleq2s 2719 . . . . . . . . 9 |- ( k e. NN -> ( seq 1 ( + , ( NN X. { A } ) ) ` ( k + 1 ) ) = ( ( seq 1 ( + , ( NN X. { A } ) ) ` k ) + ( ( NN X. { A } ) ` ( k + 1 ) ) ) )
2827adantl 482 . . . . . . . 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 11032 . . . . . . . . . 10 |- ( k e. NN -> ( k + 1 ) e. NN )
30 fvconst2g 6467 . . . . . . . . . 10 |- ( ( A e. CC /\ ( k + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
3129, 30sylan2 491 . . . . . . . . 9 |- ( ( A e. CC /\ k e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
3231oveq2d 6666 . . . . . . . 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 2656 . . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( seq 1 ( + , ( NN X. { A } ) ) ` ( k + 1 ) ) = ( ( seq 1 ( + , ( NN X. { A } ) ) ` k ) + A ) )
34 nncn 11028 . . . . . . . . 9 |- ( k e. NN -> k e. CC )
35 id 22 . . . . . . . . 9 |- ( A e. CC -> A e. CC )
36 ax-1cn 9994 . . . . . . . . . 10 |- 1 e. CC
37 adddir 10031 . . . . . . . . . 10 |- ( ( k e. CC /\ 1 e. CC /\ A e. CC ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + ( 1 x. A ) ) )
3836, 37mp3an2 1412 . . . . . . . . 9 |- ( ( k e. CC /\ A e. CC ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + ( 1 x. A ) ) )
3934, 35, 38syl2anr 495 . . . . . . . 8 |- ( ( A e. CC /\ k e. NN ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + ( 1 x. A ) ) )
4021adantr 481 . . . . . . . . 9 |- ( ( A e. CC /\ k e. NN ) -> ( 1 x. A ) = A )
4140oveq2d 6666 . . . . . . . 8 |- ( ( A e. CC /\ k e. NN ) -> ( ( k x. A ) + ( 1 x. A ) ) = ( ( k x. A ) + A ) )
4239, 41eqtrd 2656 . . . . . . 7 |- ( ( A e. CC /\ k e. NN ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + A ) )
4333, 42eqeq12d 2637 . . . . . 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 236 . . . . 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 451 . . . 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 29 . . 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 11038 . 2 |- ( N e. NN -> ( A e. CC -> ( seq 1 ( + , ( NN X. { A } ) ) ` N ) = ( N x. A ) ) )
4847impcom 446 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 384   = wceq 1483   e. wcel 1990   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   ZZ>=cuz 11687   seqcseq 12801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802
This theorem is referenced by:  fsumconst  14522  vitalilem4  23380  ovoliunnfl  33451  voliunnfl  33453
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