Theorem ser1consti 7374

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 3835	. . 3 |- (j = 1 -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` 1))
2		opreq1 4026	. . 3 |- (j = 1 -> (j x. A) = (1 x. A))
3	1, 2	eqeq12d 1532	. 2 |- (j = 1 -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` 1) = (1 x. A)))
4		fveq2 3835	. . 3 |- (j = k -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` k))
5		opreq1 4026	. . 3 |- (j = k -> (j x. A) = (k x. A))
6	4, 5	eqeq12d 1532	. 2 |- (j = k -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` k) = (k x. A)))
7		fveq2 3835	. . 3 |- (j = (k + 1) -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` (k + 1)))
8		opreq1 4026	. . 3 |- (j = (k + 1) -> (j x. A) = ((k + 1) x. A))
9	7, 8	eqeq12d 1532	. 2 |- (j = (k + 1) -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` (k + 1)) = ((k + 1) x. A)))
10		fveq2 3835	. . 3 |- (j = N -> (( + seq1 (NN X. {A}))` j) = (( + seq1 (NN X. {A}))` N))
11		opreq1 4026	. . 3 |- (j = N -> (j x. A) = (N x. A))
12	10, 11	eqeq12d 1532	. 2 |- (j = N -> ((( + seq1 (NN X. {A}))` j) = (j x. A) <-> (( + seq1 (NN X. {A}))` N) = (N x. A)))
13		1nn 6079	. . . 4 |- 1 e. NN
14		ser1const.1	. . . . . 6 |- A e. CC
15	14	elisseti 1864	. . . . 5 |- A e. V
16	15	fvconst2 3960	. . . 4 |- (1 e. NN -> ((NN X. {A})` 1) = A)
17	13, 16	ax-mp 7	. . 3 |- ((NN X. {A})` 1) = A
18		addex 5471	. . . 4 |- + e. V
19		nnex 6078	. . . . 5 |- NN e. V
20		snex 2826	. . . . 5 |- {A} e. V
21	19, 20	xpex 3349	. . . 4 |- (NN X. {A}) e. V
22	18, 21	seq11 6682	. . 3 |- (( + seq1 (NN X. {A}))` 1) = ((NN X. {A})` 1)
23	14	mulid2i 5487	. . 3 |- (1 x. A) = A
24	17, 22, 23	3eqtr4i 1548	. 2 |- (( + seq1 (NN X. {A}))` 1) = (1 x. A)
25	18, 21	seq1p1 6683	. . . . . 6 |- (k e. NN -> (( + seq1 (NN X. {A}))` (k + 1)) = ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))))
26		peano2nn 6080	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
27	15	fvconst2 3960	. . . . . . . 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 4034	. . . . . 6 |- (k e. NN -> ((( + seq1 (NN X. {A}))` k) + ((NN X. {A})` (k + 1))) = ((( + seq1 (NN X. {A}))` k) + A))
30	25, 29	eqtrd 1550	. . . . 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 4026	. . . . 5 |- ((( + seq1 (NN X. {A}))` k) = (k x. A) -> ((( + seq1 (NN X. {A}))` k) + A) = ((k x. A) + A))
33		nncn 6075	. . . . . . 7 |- (k e. NN -> k e. CC)
34		ax1cn 5423	. . . . . . . 8 |- 1 e. CC
35		adddir 5473	. . . . . . . 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 914	. . . . . . 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 4030	. . . . . 6 |- ((k x. A) + (1 x. A)) = ((k x. A) + A)
39	37, 38	syl6req 1567	. . . . 5 |- (k e. NN -> ((k x. A) + A) = ((k + 1) x. A))
40	32, 39	sylan9eqr 1572	. . . 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 1550	. . 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 371	. 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 6082	1 |- (N e. NN -> (( + seq1 (NN X. {A}))` N) = (N x. A))

Syntax hints:   -> wi 3   /\ wa 221   = wceq 992   e. wcel 994  {csn 2467   X. cxp 3249  ` cfv 3263  (class class class)co 4021  CCcc 5386  1c1 5389   + caddc 5391   x. cmul 5393  NNcn 5450   seq1 cseq1 6672

This theorem is referenced by:  ser10 7375

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-n 6070  df-n0 6268  df-z 6304  df-seq1 6673

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