Theorem ser1const 7107

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

Hypothesis

Ref	Expression
ser1const.1	|- A e. CC

Assertion

Ref	Expression
ser1const	|- (N e. NN -> (( + seq1 (NN X. {A}))` N) = (N x. A))

Proof of Theorem ser1const

Step	Hyp	Ref	Expression
1		fveq2 3709	. . 3 |- (j = 1 -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` 1))
2		opreq1 3953	. . 3 |- (j = 1 -> (j x. A) = (1 x. A))
3	1, 2	eqeq12d 1481	. 2 |- (j = 1 -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` 1) = (1 x. A)))
4		fveq2 3709	. . 3 |- (j = k -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` k))
5		opreq1 3953	. . 3 |- (j = k -> (j x. A) = (k x. A))
6	4, 5	eqeq12d 1481	. 2 |- (j = k -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` k) = (k x. A)))
7		fveq2 3709	. . 3 |- (j = (k + 1) -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` (k + 1)))
8		opreq1 3953	. . 3 |- (j = (k + 1) -> (j x. A) = ((k + 1) x. A))
9	7, 8	eqeq12d 1481	. 2 |- (j = (k + 1) -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` (k + 1)) = ((k + 1) x. A)))
10		fveq2 3709	. . 3 |- (j = N -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` N))
11		opreq1 3953	. . 3 |- (j = N -> (j x. A) = (N x. A))
12	10, 11	eqeq12d 1481	. 2 |- (j = N -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` N) = (N x. A)))
13		1nn 5882	. . . 4 |- 1 e. NN
14		ser1const.1	. . . . . 6 |- A e. CC
15	14	elisseti 1809	. . . . 5 |- A e. V
16	15	fvconst2 3831	. . . 4 |- (1 e. NN -> ((NN X. {A})` 1) = A)
17	13, 16	ax-mp 7	. . 3 |- ((NN X. {A})` 1) = A
18		addex 5289	. . . 4 |- + e. V
19		nnex 5881	. . . . 5 |- NN e. V
20		snex 2740	. . . . 5 |- {A} e. V
21	19, 20	xpex 3250	. . . 4 |- (NN X. {A}) e. V
22	18, 21	seq11 6254	. . 3 |- (( + seq1 (NN X. {A}))` 1) = ((NN X. {A})` 1)
23	14	mulid2 5305	. . 3 |- (1 x. A) = A
24	17, 22, 23	3eqtr4 1497	. 2 |- (( + seq1 (NN X. {A}))` 1) = (1 x. A)
25	18, 21	seq1p1 6255	. . . . . 6 |- (k e. NN -> (( + seq1 (NN X. {A}))` (k + 1)) = ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))))
26		peano2nn 5883	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
27	15	fvconst2 3831	. . . . . . . 8 |- ((k + 1) e. NN -> ((NN X. {A})` (k + 1)) = A)
28	26, 27	syl 10	. . . . . . 7 |- (k e. NN -> ((NN X. {A})` (k + 1)) = A)
29	28	opreq2d 3961	. . . . . 6 |- (k e. NN -> ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))) = ((( + seq1 (NN X. {A}))` k) + A))
30	25, 29	eqtrd 1499	. . . . 5 |- (k e. NN -> (( + seq1 (NN X. {A}))` (k + 1)) = ((( + seq1 (NN X. {A}))` k) + A))
31	30	adantr 389	. . . 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 3953	. . . . 5 |- ((( + seq1 (NN X. {A}))` k) = (k x. A) -> ((( + seq1 (NN X. {A}))` k) + A) = ((k x. A) + A))
33		nncnt 5878	. . . . . . 7 |- (k e. NN -> k e. CC)
34		ax1cn 5241	. . . . . . . 8 |- 1 e. CC
35		adddirt 5291	. . . . . . . 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 905	. . . . . . 7 |- (k e. CC -> ((k + 1) x. A) = ((k x. A) + (1 x. A)))
37	33, 36	syl 10	. . . . . 6 |- (k e. NN -> ((k + 1) x. A) = ((k x. A) + (1 x. A)))
38	23	opreq2i 3957	. . . . . 6 |- ((k x. A) + (1 x. A)) = ((k x. A) + A)
39	37, 38	syl6req 1516	. . . . 5 |- (k e. NN -> ((k x. A) + A) = ((k + 1) x. A))
40	32, 39	sylan9eqr 1521	. . . 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 1499	. . 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 373	. 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 5885	1 |- (N e. NN -> (( + seq1 (NN X. {A}))` N) = (N x. A))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  {csn 2399   X. cxp 3158  ` cfv 3172  (class class class)co 3948  CCcc 5204  1c1 5207   + caddc 5209   x. cmul 5211  NNcn 5268   seq1 cseq1 6244

This theorem is referenced by:  ser10 7108

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-n 5873  df-n0 6047  df-z 6083  df-seq1 6245

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