plyaddlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > plyaddlem1		Structured version Visualization version GIF version

Theorem plyaddlem1 26252

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 11236	. . . 4 ⊢ ℂ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
3		sumex 15724	. . . 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 15724	. . . 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 7717	. 2 ⊢ (𝜑 → (𝐹 ∘_f + 𝐺) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
10		fzfid 14014	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ Fin)
11		elfznn0 13660	. . . . . 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 7104	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
15		expcl 14120	. . . . . . . 8 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑧↑𝑘) ∈ ℂ)
16	15	adantll 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝑧↑𝑘) ∈ ℂ)
17	14, 16	mulcld 11281	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
18	11, 17	sylan2 593	. . . . 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 7104	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (𝐵‘𝑘) ∈ ℂ)
22	21, 16	mulcld 11281	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
23	11, 22	sylan2 593	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
24	10, 18, 23	fsumadd 15776	. . . 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 6737	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 Fn ℕ₀)
26	19	ffnd 6737	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 Fn ℕ₀)
27		nn0ex 12532	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℕ₀ ∈ V)
29		inidm 4227	. . . . . . . . . 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 7708	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝐴 ∘_f + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
33	32	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → ((𝐴 ∘_f + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
34	33	oveq1d 7446	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) + (𝐵‘𝑘)) · (𝑧↑𝑘)))
35	14, 21, 16	adddird 11286	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (((𝐴‘𝑘) + (𝐵‘𝑘)) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))))
36	34, 35	eqtrd 2777	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ₀) → (((𝐴 ∘_f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))))
37	11, 36	sylan2 593	. . . . 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 15738	. . . 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 12639	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
41		plyaddlem.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
42	41, 39	ifcld 4572	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ₀)
43	42	nn0zd 12639	. . . . . . . . 9 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ)
44	39	nn0red 12588	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
45	41	nn0red 12588	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
46		max1 13227	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
47	44, 45, 46	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
48		eluz2 12884	. . . . . . . . 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 1344	. . . . . . . 8 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑀))
50		fzss2 13604	. . . . . . . 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 13660	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ₀)
54	53, 17	sylan2 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
55		eldifn 4132	. . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . . 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 12920	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ₀ = (ℤ_≥‘0)
61		peano2nn0 12566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 + 1) ∈ ℕ₀)
62	39, 61	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
63	62, 60	eleqtrdi 2851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ_≥‘0))
64		uzsplit 13636	. . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ₀ = ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))))
67	39	nn0cnd 12589	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℂ)
68		ax-1cn 11213	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
69		pncan 11514	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
70	67, 68, 69	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
71	70	oveq2d 7447	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0...((𝑀 + 1) − 1)) = (0...𝑀))
72	71	uneq1d 4167	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1))) = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
73	66, 72	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ₀ = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
74	73	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ℕ₀ = ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
75	59, 74	eleqtrd 2843	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → 𝑘 ∈ ((0...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
76		elun 4153	. . . . . . . . . . . . . 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 6740	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐴)
81		ssun2 4179	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘(𝑀 + 1)) ⊆ ((0...((𝑀 + 1) − 1)) ∪ (ℤ_≥‘(𝑀 + 1)))
82	81, 66	sseqtrrid 4027	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ ℕ₀)
83	12	fdmd 6746	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐴 = ℕ₀)
84	82, 83	sseqtrrd 4021	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ dom 𝐴)
85		funfvima2 7251	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐴 ∧ (ℤ_≥‘(𝑀 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1)))))
86	80, 84, 85	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ_≥‘(𝑀 + 1)))))
87	86	ad2antrr 726	. . . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
91	88, 90	eleqtrd 2843	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ {0})
92		elsni 4643	. . . . . . . . 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 7446	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
95	58, 16	sylan2 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
96	95	mul02d 11459	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
97	94, 96	eqtrd 2777	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
98	52, 54, 97, 10	fsumss 15761	. . . . 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 12639	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
100		max2 13229	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
101	44, 45, 100	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
102		eluz2 12884	. . . . . . . . 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 1344	. . . . . . . 8 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ_≥‘𝑁))
104		fzss2 13604	. . . . . . . 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 13660	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
108	107, 22	sylan2 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
109		eldifn 4132	. . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . . 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 12566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
115	41, 114	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
116	115, 60	eleqtrdi 2851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘0))
117		uzsplit 13636	. . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ₀ = ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))))
120	41	nn0cnd 12589	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℂ)
121		pncan 11514	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
122	120, 68, 121	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
123	122	oveq2d 7447	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
124	123	uneq1d 4167	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
125	119, 124	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ₀ = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
126	125	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ℕ₀ = ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
127	113, 126	eleqtrd 2843	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → 𝑘 ∈ ((0...𝑁) ∪ (ℤ_≥‘(𝑁 + 1))))
128		elun 4153	. . . . . . . . . . . . . 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 6740	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐵)
133		ssun2 4179	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘(𝑁 + 1)) ⊆ ((0...((𝑁 + 1) − 1)) ∪ (ℤ_≥‘(𝑁 + 1)))
134	133, 119	sseqtrrid 4027	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ_≥‘(𝑁 + 1)) ⊆ ℕ₀)
135	19	fdmd 6746	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐵 = ℕ₀)
136	134, 135	sseqtrrd 4021	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ_≥‘(𝑁 + 1)) ⊆ dom 𝐵)
137		funfvima2 7251	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ (ℤ_≥‘(𝑁 + 1)) ⊆ dom 𝐵) → (𝑘 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝐵‘𝑘) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1)))))
138	132, 136, 137	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝐵‘𝑘) ∈ (𝐵 “ (ℤ_≥‘(𝑁 + 1)))))
139	138	ad2antrr 726	. . . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
143	140, 142	eleqtrd 2843	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝐵‘𝑘) ∈ {0})
144		elsni 4643	. . . . . . . . 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 7446	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
147	112, 16	sylan2 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝑧↑𝑘) ∈ ℂ)
148	147	mul02d 11459	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑘)) = 0)
149	146, 148	eqtrd 2777	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = 0)
150	106, 108, 149, 10	fsumss 15761	. . . . 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 7449	. . . 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 2787	. . 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 5242	. 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 2780	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 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ifcif 4525 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 “ cima 5688 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘_f cof 7695 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547 ↑cexp 14102 Σcsu 15722 Polycply 26223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723

This theorem is referenced by: plyaddlem 26254 coeaddlem 26288

W3C validator