Theorem ser1const 13235

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 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))

Proof of Theorem ser1const

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6493	. . . . 5 ⊢ (𝑗 = 1 → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘1))
2		oveq1 6977	. . . . 5 ⊢ (𝑗 = 1 → (𝑗 · 𝐴) = (1 · 𝐴))
3	1, 2	eqeq12d 2787	. . . 4 ⊢ (𝑗 = 1 → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘1) = (1 · 𝐴)))
4	3	imbi2d 333	. . 3 ⊢ (𝑗 = 1 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘1) = (1 · 𝐴))))
5		fveq2 6493	. . . . 5 ⊢ (𝑗 = 𝑘 → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘𝑘))
6		oveq1 6977	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑗 · 𝐴) = (𝑘 · 𝐴))
7	5, 6	eqeq12d 2787	. . . 4 ⊢ (𝑗 = 𝑘 → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴)))
8	7	imbi2d 333	. . 3 ⊢ (𝑗 = 𝑘 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴))))
9		fveq2 6493	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)))
10		oveq1 6977	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 · 𝐴) = ((𝑘 + 1) · 𝐴))
11	9, 10	eqeq12d 2787	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴)))
12	11	imbi2d 333	. . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴))))
13		fveq2 6493	. . . . 5 ⊢ (𝑗 = 𝑁 → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘𝑁))
14		oveq1 6977	. . . . 5 ⊢ (𝑗 = 𝑁 → (𝑗 · 𝐴) = (𝑁 · 𝐴))
15	13, 14	eqeq12d 2787	. . . 4 ⊢ (𝑗 = 𝑁 → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴)))
16	15	imbi2d 333	. . 3 ⊢ (𝑗 = 𝑁 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))))
17		1z 11819	. . . 4 ⊢ 1 ∈ ℤ
18		1nn 11446	. . . . . 6 ⊢ 1 ∈ ℕ
19		fvconst2g 6785	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℕ) → ((ℕ × {𝐴})‘1) = 𝐴)
20	18, 19	mpan2 678	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((ℕ × {𝐴})‘1) = 𝐴)
21		mulid2 10432	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
22	20, 21	eqtr4d 2811	. . . 4 ⊢ (𝐴 ∈ ℂ → ((ℕ × {𝐴})‘1) = (1 · 𝐴))
23	17, 22	seq1i 13192	. . 3 ⊢ (𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘1) = (1 · 𝐴))
24		oveq1 6977	. . . . . 6 ⊢ ((seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴) = ((𝑘 · 𝐴) + 𝐴))
25		seqp1 13193	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1))))
26		nnuz 12089	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
27	25, 26	eleq2s 2878	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1))))
28	27	adantl 474	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1))))
29		peano2nn 11447	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
30		fvconst2g 6785	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ) → ((ℕ × {𝐴})‘(𝑘 + 1)) = 𝐴)
31	29, 30	sylan2 583	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((ℕ × {𝐴})‘(𝑘 + 1)) = 𝐴)
32	31	oveq2d 6986	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1))) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴))
33	28, 32	eqtrd 2808	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴))
34		nncn 11442	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
35		id 22	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
36		ax-1cn 10387	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
37		adddir 10424	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + (1 · 𝐴)))
38	36, 37	mp3an2 1428	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + (1 · 𝐴)))
39	34, 35, 38	syl2anr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + (1 · 𝐴)))
40	21	adantr 473	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (1 · 𝐴) = 𝐴)
41	40	oveq2d 6986	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 · 𝐴) + (1 · 𝐴)) = ((𝑘 · 𝐴) + 𝐴))
42	39, 41	eqtrd 2808	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + 𝐴))
43	33, 42	eqeq12d 2787	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴) ↔ ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴) = ((𝑘 · 𝐴) + 𝐴)))
44	24, 43	syl5ibr 238	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴)))
45	44	expcom 406	. . . 4 ⊢ (𝑘 ∈ ℕ → (𝐴 ∈ ℂ → ((seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴))))
46	45	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴)) → (𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴))))
47	4, 8, 12, 16, 23, 46	nnind 11453	. 2 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ ℂ → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴)))
48	47	impcom 399	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {csn 4435   × cxp 5399  ‘cfv 6182  (class class class)co 6970  ℂcc 10327  1c1 10330   + caddc 10332   · cmul 10334  ℕcn 11433  ℤ_≥cuz 12052  seqcseq 13178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-2nd 7496  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-er 8083  df-en 8301  df-dom 8302  df-sdom 8303  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-n0 11702  df-z 11788  df-uz 12053  df-seq 13179

This theorem is referenced by:  fsumconst  14999  vitalilem4  23909  ovoliunnfl  34375  voliunnfl  34377