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Theorem ply1termlem 25708

Description: Lemma for ply1term 25709. (Contributed by Mario Carneiro, 26-Jul-2014.)

**Hypothesis**
Ref	Expression
ply1term.1	⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))

**Assertion**
Ref	Expression
ply1termlem	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))

Distinct variable groups: 𝑧,𝑘,𝐴 𝑘,𝑁,𝑧 Allowed substitution hints: 𝐹(𝑧,𝑘)

**Proof of Theorem ply1termlem**
Step	Hyp	Ref	Expression
1		ply1term.1	. 2 ⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))
2		simplr 767	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℕ₀)
3		nn0uz 12860	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2843	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ (ℤ_≥‘0))
5		fzss1 13536	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (𝑁...𝑁) ⊆ (0...𝑁))
6	4, 5	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (𝑁...𝑁) ⊆ (0...𝑁))
7		elfz1eq 13508	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑁...𝑁) → 𝑘 = 𝑁)
8	7	adantl 482	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑘 = 𝑁)
9		iftrue 4533	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → if(𝑘 = 𝑁, 𝐴, 0) = 𝐴)
10	8, 9	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → if(𝑘 = 𝑁, 𝐴, 0) = 𝐴)
11		simpll 765	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝐴 ∈ ℂ)
12	11	adantr 481	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → 𝐴 ∈ ℂ)
13	10, 12	eqeltrd 2833	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
14		simplr 767	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑧 ∈ ℂ)
15	2	adantr 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑁 ∈ ℕ₀)
16	8, 15	eqeltrd 2833	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑘 ∈ ℕ₀)
17	14, 16	expcld 14107	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → (𝑧↑𝑘) ∈ ℂ)
18	13, 17	mulcld 11230	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (𝑁...𝑁)) → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) ∈ ℂ)
19		eldifn 4126	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁)) → ¬ 𝑘 ∈ (𝑁...𝑁))
20	19	adantl 482	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → ¬ 𝑘 ∈ (𝑁...𝑁))
21	2	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → 𝑁 ∈ ℕ₀)
22	21	nn0zd 12580	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → 𝑁 ∈ ℤ)
23		fzsn 13539	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
24	23	eleq2d 2819	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 ∈ {𝑁}))
25		elsn2g 4665	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ {𝑁} ↔ 𝑘 = 𝑁))
26	24, 25	bitrd 278	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 = 𝑁))
27	22, 26	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 = 𝑁))
28	20, 27	mtbid 323	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → ¬ 𝑘 = 𝑁)
29	28	iffalsed 4538	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → if(𝑘 = 𝑁, 𝐴, 0) = 0)
30	29	oveq1d 7420	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
31		simpr 485	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
32		eldifi 4125	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁)) → 𝑘 ∈ (0...𝑁))
33		elfznn0 13590	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
34	32, 33	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁)) → 𝑘 ∈ ℕ₀)
35		expcl 14041	. . . . . . . 8 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑧↑𝑘) ∈ ℂ)
36	31, 34, 35	syl2an 596	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (𝑧↑𝑘) ∈ ℂ)
37	36	mul02d 11408	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (0 · (𝑧↑𝑘)) = 0)
38	30, 37	eqtrd 2772	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = 0)
39		fzfid 13934	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin)
40	6, 18, 38, 39	fsumss 15667	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (𝑁...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)))
41	2	nn0zd 12580	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℤ)
42	31, 2	expcld 14107	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (𝑧↑𝑁) ∈ ℂ)
43	11, 42	mulcld 11230	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑𝑁)) ∈ ℂ)
44		oveq2 7413	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (𝑧↑𝑘) = (𝑧↑𝑁))
45	9, 44	oveq12d 7423	. . . . . 6 ⊢ (𝑘 = 𝑁 → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁)))
46	45	fsum1 15689	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ (𝐴 · (𝑧↑𝑁)) ∈ ℂ) → Σ𝑘 ∈ (𝑁...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁)))
47	41, 43, 46	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (𝑁...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁)))
48	40, 47	eqtr3d 2774	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁)))
49	48	mpteq2dva 5247	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁))))
50	1, 49	eqtr4id 2791	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3944 ⊆ wss 3947 ifcif 4527 {csn 4627 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 · cmul 11111 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ↑cexp 14023 Σcsu 15628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629

This theorem is referenced by: ply1term 25709 coe1termlem 25763

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