Theorem ser1consti 8431

Description: Value of the partial series sum of a constant function.

Hypothesis

Ref	Expression
ser1const.1	|- A e. CC

Assertion

Ref	Expression
ser1consti	|- (N e. NN -> (( + seq1 (NN X. {A}))` N) = (N x. A))

Proof of Theorem ser1consti

Step	Hyp	Ref	Expression
1		fveq2 4681	. . 3 |- (j = 1 -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` 1))
2		opreq1 4889	. . 3 |- (j = 1 -> (j x. A) = (1 x. A))
3	1, 2	eqeq12d 1899	. 2 |- (j = 1 -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` 1) = (1 x. A)))
4		fveq2 4681	. . 3 |- (j = k -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` k))
5		opreq1 4889	. . 3 |- (j = k -> (j x. A) = (k x. A))
6	4, 5	eqeq12d 1899	. 2 |- (j = k -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` k) = (k x. A)))
7		fveq2 4681	. . 3 |- (j = (k + 1) -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` (k + 1)))
8		opreq1 4889	. . 3 |- (j = (k + 1) -> (j x. A) = ((k + 1) x. A))
9	7, 8	eqeq12d 1899	. 2 |- (j = (k + 1) -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` (k + 1)) = ((k + 1) x. A)))
10		fveq2 4681	. . 3 |- (j = N -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` N))
11		opreq1 4889	. . 3 |- (j = N -> (j x. A) = (N x. A))
12	10, 11	eqeq12d 1899	. 2 |- (j = N -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` N) = (N x. A)))
13		1nn 7117	. . . 4 |- 1 e. NN
14		ser1const.1	. . . . . 6 |- A e. CC
15	14	elisseti 2301	. . . . 5 |- A e. _V
16	15	fvconst2 4822	. . . 4 |- (1 e. NN -> ((NN X. {A})` 1) = A)
17	13, 16	ax-mp 7	. . 3 |- ((NN X. {A})` 1) = A
18		addex 6470	. . . 4 |- + e. _V
19		nnex 7116	. . . . 5 |- NN e. _V
20		snex 3492	. . . . 5 |- {A} e. _V
21	19, 20	xpex 4096	. . . 4 |- (NN X. {A}) e. _V
22	18, 21	seq11 7730	. . 3 |- (( + seq1 (NN X. {A}))` 1) = ((NN X. {A})` 1)
23	14	mulid2i 6486	. . 3 |- (1 x. A) = A
24	17, 22, 23	3eqtr4i 1921	. 2 |- (( + seq1 (NN X. {A}))` 1) = (1 x. A)
25	18, 21	seq1p1 7731	. . . . . 6 |- (k e. NN -> (( + seq1 (NN X. {A}))` (k + 1)) = ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))))
26		peano2nn 7118	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
27	15	fvconst2 4822	. . . . . . . 8 |- ((k + 1) e. NN -> ((NN X. {A})` (k + 1)) = A)
28	26, 27	syl 12	. . . . . . 7 |- (k e. NN -> ((NN X. {A})` (k + 1)) = A)
29	28	opreq2d 4898	. . . . . 6 |- (k e. NN -> ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))) = ((( + seq1 (NN X. {A}))` k) + A))
30	25, 29	eqtrd 1925	. . . . 5 |- (k e. NN -> (( + seq1 (NN X. {A}))` (k + 1)) = ((( + seq1 (NN X. {A}))` k) + A))
31	30	adantr 425	. . . 4 |- ((k e. NN /\ (( + seq1 (NN X. {A}))` k) = (k x. A)) -> (( + seq1 (NN X. {A}))` (k + 1)) = ((( + seq1 (NN X. {A}))` k) + A))
32		opreq1 4889	. . . . 5 |- ((( + seq1 (NN X. {A}))` k) = (k x. A) -> ((( + seq1 (NN X. {A}))` k) + A) = ((k x. A) + A))
33		nncn 7113	. . . . . . 7 |- (k e. NN -> k e. CC)
34		ax1cn 6422	. . . . . . . 8 |- 1 e. CC
35		adddir 6472	. . . . . . . 8 |- ((k e. CC /\ 1 e. CC /\ A e. CC) -> ((k + 1) x. A) = ((k x. A) + (1 x. A)))
36	34, 14, 35	mp3an23 1183	. . . . . . 7 |- (k e. CC -> ((k + 1) x. A) = ((k x. A) + (1 x. A)))
37	33, 36	syl 12	. . . . . 6 |- (k e. NN -> ((k + 1) x. A) = ((k x. A) + (1 x. A)))
38	23	opreq2i 4893	. . . . . 6 |- ((k x. A) + (1 x. A)) = ((k x. A) + A)
39	37, 38	syl6req 1945	. . . . 5 |- (k e. NN -> ((k x. A) + A) = ((k + 1) x. A))
40	32, 39	sylan9eqr 1951	. . . 4 |- ((k e. NN /\ (( + seq1 (NN X. {A}))` k) = (k x. A)) -> ((( + seq1 (NN X. {A}))` k) + A) = ((k + 1) x. A))
41	31, 40	eqtrd 1925	. . 3 |- ((k e. NN /\ (( + seq1 (NN X. {A}))` k) = (k x. A)) -> (( + seq1 (NN X. {A}))` (k + 1)) = ((k + 1) x. A))
42	41	ex 402	. 2 |- (k e. NN -> ((( + seq1 (NN X. {A}))` k) = (k x. A) -> (( + seq1 (NN X. {A}))` (k + 1)) = ((k + 1) x. A)))
43	3, 6, 9, 12, 24, 42	nnind 7120	1 |- (N e. NN -> (( + seq1 (NN X. {A}))` N) = (N x. A))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {csn 3044   X. cxp 3984  ` cfv 3998  (class class class)co 4884  CCcc 6384  1c1 6387   + caddc 6389   x. cmul 6391  NNcn 6449   seq1 cseq1 7720

This theorem is referenced by:  ser10 8432

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-seq1 7721

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