Theorem ser1const 7124

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 3719	. . 3 |- (j = 1 -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` 1))
2		opreq1 3963	. . 3 |- (j = 1 -> (j x. A) = (1 x. A))
3	1, 2	eqeq12d 1487	. 2 |- (j = 1 -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` 1) = (1 x. A)))
4		fveq2 3719	. . 3 |- (j = k -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` k))
5		opreq1 3963	. . 3 |- (j = k -> (j x. A) = (k x. A))
6	4, 5	eqeq12d 1487	. 2 |- (j = k -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` k) = (k x. A)))
7		fveq2 3719	. . 3 |- (j = (k + 1) -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` (k + 1)))
8		opreq1 3963	. . 3 |- (j = (k + 1) -> (j x. A) = ((k + 1) x. A))
9	7, 8	eqeq12d 1487	. 2 |- (j = (k + 1) -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` (k + 1)) = ((k + 1) x. A)))
10		fveq2 3719	. . 3 |- (j = N -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` N))
11		opreq1 3963	. . 3 |- (j = N -> (j x. A) = (N x. A))
12	10, 11	eqeq12d 1487	. 2 |- (j = N -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` N) = (N x. A)))
13		1nn 5892	. . . 4 |- 1 e. NN
14		ser1const.1	. . . . . 6 |- A e. CC
15	14	elisseti 1815	. . . . 5 |- A e. V
16	15	fvconst2 3841	. . . 4 |- (1 e. NN -> ((NN X. {A})` 1) = A)
17	13, 16	ax-mp 7	. . 3 |- ((NN X. {A})` 1) = A
18		addex 5300	. . . 4 |- + e. V
19		nnex 5891	. . . . 5 |- NN e. V
20		snex 2746	. . . . 5 |- {A} e. V
21	19, 20	xpex 3256	. . . 4 |- (NN X. {A}) e. V
22	18, 21	seq11 6267	. . 3 |- (( + seq1 (NN X. {A}))` 1) = ((NN X. {A})` 1)
23	14	mulid2 5316	. . 3 |- (1 x. A) = A
24	17, 22, 23	3eqtr4 1503	. 2 |- (( + seq1 (NN X. {A}))` 1) = (1 x. A)
25	18, 21	seq1p1 6268	. . . . . 6 |- (k e. NN -> (( + seq1 (NN X. {A}))` (k + 1)) = ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))))
26		peano2nn 5893	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
27	15	fvconst2 3841	. . . . . . . 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 3971	. . . . . 6 |- (k e. NN -> ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))) = ((( + seq1 (NN X. {A}))` k) + A))
30	25, 29	eqtrd 1505	. . . . 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 3963	. . . . 5 |- ((( + seq1 (NN X. {A}))` k) = (k x. A) -> ((( + seq1 (NN X. {A}))` k) + A) = ((k x. A) + A))
33		nncnt 5888	. . . . . . 7 |- (k e. NN -> k e. CC)
34		ax1cn 5252	. . . . . . . 8 |- 1 e. CC
35		adddirt 5302	. . . . . . . 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 907	. . . . . . 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 3967	. . . . . 6 |- ((k x. A) + (1 x. A)) = ((k x. A) + A)
39	37, 38	syl6req 1522	. . . . 5 |- (k e. NN -> ((k x. A) + A) = ((k + 1) x. A))
40	32, 39	sylan9eqr 1527	. . . 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 1505	. . 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 5895	1 |- (N e. NN -> (( + seq1 (NN X. {A}))` N) = (N x. A))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  {csn 2406   X. cxp 3164  ` cfv 3178  (class class class)co 3958  CCcc 5215  1c1 5218   + caddc 5220   x. cmul 5222  NNcn 5279   seq1 cseq1 6257

This theorem is referenced by:  ser10 7125

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-n 5883  df-n0 6057  df-z 6093  df-seq1 6258

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