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Theorem plyaddlem1 26191

Description: Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)

**Hypotheses**
Ref	Expression
plyaddlem.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
plyaddlem.2	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
plyaddlem.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
plyaddlem.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
plyaddlem.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
plyaddlem.b	⊢ (𝜑 → 𝐵:ℕ₀⟶ℂ)
plyaddlem.a2	⊢ (𝜑 → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
plyaddlem.b2	⊢ (𝜑 → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
plyaddlem.f	⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
plyaddlem.g	⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))

**Assertion**
Ref	Expression
plyaddlem1	⊢ (𝜑 → (𝐹 ∘_f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘))))

Distinct variable groups: 𝐵,𝑘 𝑘,𝑀 𝑘,𝑁 𝑧,𝑘,𝜑 Allowed substitution hints: 𝐴(𝑧,𝑘) 𝐵(𝑧) 𝑆(𝑧,𝑘) 𝐹(𝑧,𝑘) 𝐺(𝑧,𝑘) 𝑀(𝑧) 𝑁(𝑧)

**Proof of Theorem plyaddlem1**
Step	Hyp	Ref	Expression
1		cnex 11121	. . . 4 ⊢ ℂ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
3		sumex 15625	. . . 4 ⊢ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V)
5		sumex 15625	. . . 4 ⊢ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V)
7		plyaddlem.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
8		plyaddlem.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
9	2, 4, 6, 7, 8	offval2 7654	. 2 ⊢ (𝜑 → (𝐹 ∘_f + 𝐺) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
10		fzfid 13910	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ Fin)
11		elfznn0 13550	. . . . . 6 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ₀)
12		plyaddlem.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ₀⟶ℂ)
14	13	ffvelcdmda 7040	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
15		expcl 14016	. . . . . . . 8 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑧↑𝑘) ∈ ℂ)
16	15	adantll 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝑧↑𝑘) ∈ ℂ)
17	14, 16	mulcld 11166	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
18	11, 17	sylan2 594	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
19		plyaddlem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵:ℕ₀⟶ℂ)
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐵:ℕ₀⟶ℂ)
21	20	ffvelcdmda 7040	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐵‘𝑘) ∈ ℂ)
22	21, 16	mulcld 11166	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
23	11, 22	sylan2 594	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
24	10, 18, 23	fsumadd 15677	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))) = (Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))((𝐵‘𝑘) · (𝑧↑𝑘))))
25	12	ffnd 6673	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 Fn ℕ₀)
26	19	ffnd 6673	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 Fn ℕ₀)
27		nn0ex 12421	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℕ₀ ∈ V)
29		inidm 4181	. . . . . . . . . 10 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
30		eqidd 2738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) = (𝐴‘𝑘))
31		eqidd 2738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐵‘𝑘) = (𝐵‘𝑘))
32	25, 26, 28, 28, 29, 30, 31	ofval 7645	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴 ∘_f + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
33	32	adantlr 716	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴 ∘_f + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
34	33	oveq1d 7385	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) + (𝐵‘𝑘)) · (𝑧↑𝑘)))
35	14, 21, 16	adddird 11171	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (((𝐴‘𝑘) + (𝐵‘𝑘)) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))))
36	34, 35	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))))
37	11, 36	sylan2 594	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → (((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))))
38	37	sumeq2dv 15639	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))))
39		plyaddlem.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
40	39	nn0zd 12527	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
41		plyaddlem.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
42	41, 39	ifcld 4528	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ₀)
43	42	nn0zd 12527	. . . . . . . . 9 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ)
44	39	nn0red 12477	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
45	41	nn0red 12477	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
46		max1 13114	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
47	44, 45, 46	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
48		eluz2 12771	. . . . . . . . 9 ⊢ (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ ∧ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
49	40, 43, 47, 48	syl3anbrc 1345	. . . . . . . 8 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑀))
50		fzss2 13494	. . . . . . . 8 ⊢ (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑀) → (0...𝑀) ⊆ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
51	49, 50	syl 17	. . . . . . 7 ⊢ (𝜑 → (0...𝑀) ⊆ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
52	51	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ⊆ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
53		elfznn0 13550	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ₀)
54	53, 17	sylan2 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
55		eldifn 4086	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀)) → ¬ 𝑘 ∈ (0...𝑀))
56	55	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ¬ 𝑘 ∈ (0...𝑀))
57		eldifi 4085	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀)) → 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
58	57, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀)) → 𝑘 ∈ ℕ₀)
59	58	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → 𝑘 ∈ ℕ₀)
60		nn0uz 12803	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ₀ = (ℤ_≥‘0)
61		peano2nn0 12455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 + 1) ∈ ℕ₀)
62	39, 61	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
63	62, 60	eleqtrdi 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ_≥‘0))
64		uzsplit 13526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘0) → (ℤ_≥‘0) = ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))))
65	63, 64	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ_≥‘0) = ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))))
66	60, 65	eqtrid 2784	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ₀ = ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))))
67	39	nn0cnd 12478	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℂ)
68		ax-1cn 11098	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
69		pncan 11400	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
70	67, 68, 69	sylancl 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
71	70	oveq2d 7386	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0...((𝑀 + 1) − 1)) = (0...𝑀))
72	71	uneq1d 4121	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))) = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
73	66, 72	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ₀ = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
74	73	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ℕ₀ = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
75	59, 74	eleqtrd 2839	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → 𝑘 ∈ ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
76		elun 4107	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))) ↔ (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ_≥‘(𝑀 + 1))))
77	75, 76	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝑘 ∈ (0...𝑀) ∨ 𝑘 ∈ (ℤ_≥‘(𝑀 + 1))))
78	77	ord 865	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (¬ 𝑘 ∈ (0...𝑀) → 𝑘 ∈ (ℤ_≥‘(𝑀 + 1))))
79	56, 78	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → 𝑘 ∈ (ℤ_≥‘(𝑀 + 1)))
80	12	ffund 6676	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐴)
81		ssun2 4133	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘(𝑀 + 1)) ⊆ ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1)))
82	81, 66	sseqtrrid 3979	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ ℕ₀)
83	12	fdmd 6682	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐴 = ℕ₀)
84	82, 83	sseqtrrd 3973	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ dom 𝐴)
85		funfvima2 7189	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐴 ∧ (ℤ_≥‘(𝑀 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1)))))
86	80, 84, 85	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1)))))
87	86	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝑘 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1)))))
88	79, 87	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1))))
89		plyaddlem.a2	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
90	89	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
91	88, 90	eleqtrd 2839	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ {0})
92		elsni 4599	. . . . . . . . 9 ⊢ ((𝐴‘𝑘) ∈ {0} → (𝐴‘𝑘) = 0)
93	91, 92	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝐴‘𝑘) = 0)
94	93	oveq1d 7385	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
95	58, 16	sylan2 594	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
96	95	mul02d 11345	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
97	94, 96	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
98	52, 54, 97, 10	fsumss 15662	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))((𝐴‘𝑘) · (𝑧↑𝑘)))
99	41	nn0zd 12527	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
100		max2 13116	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
101	44, 45, 100	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
102		eluz2 12771	. . . . . . . . 9 ⊢ (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ ∧ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
103	99, 43, 101, 102	syl3anbrc 1345	. . . . . . . 8 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑁))
104		fzss2 13494	. . . . . . . 8 ⊢ (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑁) → (0...𝑁) ⊆ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
105	103, 104	syl 17	. . . . . . 7 ⊢ (𝜑 → (0...𝑁) ⊆ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
106	105	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ⊆ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
107		elfznn0 13550	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
108	107, 22	sylan2 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
109		eldifn 4086	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁)) → ¬ 𝑘 ∈ (0...𝑁))
110	109	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ¬ 𝑘 ∈ (0...𝑁))
111		eldifi 4085	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁)) → 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
112	111, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁)) → 𝑘 ∈ ℕ₀)
113	112	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → 𝑘 ∈ ℕ₀)
114		peano2nn0 12455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
115	41, 114	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
116	115, 60	eleqtrdi 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘0))
117		uzsplit 13526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘0) → (ℤ_≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
118	116, 117	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ_≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
119	60, 118	eqtrid 2784	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ₀ = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
120	41	nn0cnd 12478	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℂ)
121		pncan 11400	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
122	120, 68, 121	sylancl 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
123	122	oveq2d 7386	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
124	123	uneq1d 4121	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
125	119, 124	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ₀ = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
126	125	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ℕ₀ = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
127	113, 126	eleqtrd 2839	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → 𝑘 ∈ ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
128		elun 4107	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))) ↔ (𝑘 ∈ (0...𝑁) ∨ 𝑘 ∈ (ℤ_≥‘(𝑁 + 1))))
129	127, 128	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝑘 ∈ (0...𝑁) ∨ 𝑘 ∈ (ℤ_≥‘(𝑁 + 1))))
130	129	ord 865	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (¬ 𝑘 ∈ (0...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝑁 + 1))))
131	110, 130	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → 𝑘 ∈ (ℤ_≥‘(𝑁 + 1)))
132	19	ffund 6676	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐵)
133		ssun2 4133	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘(𝑁 + 1)) ⊆ ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1)))
134	133, 119	sseqtrrid 3979	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ_≥‘(𝑁 + 1)) ⊆ ℕ₀)
135	19	fdmd 6682	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐵 = ℕ₀)
136	134, 135	sseqtrrd 3973	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ_≥‘(𝑁 + 1)) ⊆ dom 𝐵)
137		funfvima2 7189	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ (ℤ_≥‘(𝑁 + 1)) ⊆ dom 𝐵) → (𝑘 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝐵‘𝑘) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1)))))
138	132, 136, 137	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝐵‘𝑘) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1)))))
139	138	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝑘 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝐵‘𝑘) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1)))))
140	131, 139	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝐵‘𝑘) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1))))
141		plyaddlem.b2	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
142	141	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
143	140, 142	eleqtrd 2839	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝐵‘𝑘) ∈ {0})
144		elsni 4599	. . . . . . . . 9 ⊢ ((𝐵‘𝑘) ∈ {0} → (𝐵‘𝑘) = 0)
145	143, 144	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝐵‘𝑘) = 0)
146	145	oveq1d 7385	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
147	112, 16	sylan2 594	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝑧↑𝑘) ∈ ℂ)
148	147	mul02d 11345	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑘)) = 0)
149	146, 148	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = 0)
150	106, 108, 149, 10	fsumss 15662	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))((𝐵‘𝑘) · (𝑧↑𝑘)))
151	98, 150	oveq12d 7388	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))) = (Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))((𝐵‘𝑘) · (𝑧↑𝑘))))
152	24, 38, 151	3eqtr4d 2782	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
153	152	mpteq2dva 5193	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
154	9, 153	eqtr4d 2775	1 ⊢ (𝜑 → (𝐹 ∘_f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ∖ cdif 3900 ∪ cun 3901 ⊆ wss 3903 ifcif 4481 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5634 “ cima 5637 Fun wfun 6496 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ∘_f cof 7632 ℂcc 11038 ℝcr 11039 0cc0 11040 1c1 11041 + caddc 11043 · cmul 11045 ≤ cle 11181 − cmin 11378 ℕ₀cn0 12415 ℤcz 12502 ℤ_≥cuz 12765 ...cfz 13437 ↑cexp 13998 Σcsu 15623 Polycply 26162

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-inf2 9564 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-sup 9359 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-n0 12416 df-z 12503 df-uz 12766 df-rp 12920 df-fz 13438 df-fzo 13585 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-clim 15425 df-sum 15624

This theorem is referenced by: plyaddlem 26193 coeaddlem 26227

W3C validator