Theorem plyaddlem1 26151

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 11125	. . . 4 ⊢ ℂ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
3		sumex 15630	. . . 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 15630	. . . 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 7653	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) + Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))))
10		fzfid 13914	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ Fin)
11		elfznn0 13557	. . . . . 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 7038	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
15		expcl 14020	. . . . . . . 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 11170	. . . . . 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 7038	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) ∈ ℂ)
22	21, 16	mulcld 11170	. . . . . 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 15682	. . . 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 6671	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 Fn ℕ0)
26	19	ffnd 6671	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 Fn ℕ0)
27		nn0ex 12424	. . . . . . . . . . 11 ⊢ ℕ0 ∈ V
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℕ0 ∈ V)
29		inidm 4186	. . . . . . . . . 10 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
30		eqidd 2730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
31		eqidd 2730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
32	25, 26, 28, 28, 29, 30, 31	ofval 7644	. . . . . . . . 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 7384	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) + (𝐵‘𝑘)) · (𝑧↑𝑘)))
35	14, 21, 16	adddird 11175	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) + (𝐵‘𝑘)) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) + ((𝐵‘𝑘) · (𝑧↑𝑘))))
36	34, 35	eqtrd 2764	. . . . . 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 15644	. . . 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 12531	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
41		plyaddlem.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
42	41, 39	ifcld 4531	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
43	42	nn0zd 12531	. . . . . . . . 9 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ)
44	39	nn0red 12480	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
45	41	nn0red 12480	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
46		max1 13121	. . . . . . . . . 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 12775	. . . . . . . . 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 13501	. . . . . . . 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 13557	. . . . . . 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 4091	. . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . . 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 12811	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ0 = (ℤ≥‘0)
61		peano2nn0 12458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0)
62	39, 61	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ0)
63	62, 60	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘0))
64		uzsplit 13533	. . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ0 = ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))))
67	39	nn0cnd 12481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℂ)
68		ax-1cn 11102	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
69		pncan 11403	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀)
70	67, 68, 69	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
71	70	oveq2d 7385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0...((𝑀 + 1) − 1)) = (0...𝑀))
72	71	uneq1d 4126	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1))) = ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
73	66, 72	eqtrd 2764	. . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → 𝑘 ∈ ((0...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
76		elun 4112	. . . . . . . . . . . . . 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 864	. . . . . . . . . . . 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 6674	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐴)
81		ssun2 4138	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘(𝑀 + 1)) ⊆ ((0...((𝑀 + 1) − 1)) ∪ (ℤ≥‘(𝑀 + 1)))
82	81, 66	sseqtrrid 3987	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ0)
83	12	fdmd 6680	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐴 = ℕ0)
84	82, 83	sseqtrrd 3981	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ dom 𝐴)
85		funfvima2 7187	. . . . . . . . . . . . 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 2830	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (𝐴‘𝑘) ∈ {0})
92		elsni 4602	. . . . . . . . 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 7384	. . . . . . 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 11348	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
97	94, 96	eqtrd 2764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
98	52, 54, 97, 10	fsumss 15667	. . . . 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 12531	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
100		max2 13123	. . . . . . . . . 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 12775	. . . . . . . . 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 13501	. . . . . . . 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 13557	. . . . . . 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 4091	. . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . . 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 12458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
115	41, 114	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
116	115, 60	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘0))
117		uzsplit 13533	. . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))))
120	41	nn0cnd 12481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℂ)
121		pncan 11403	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
122	120, 68, 121	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
123	122	oveq2d 7385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
124	123	uneq1d 4126	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
125	119, 124	eqtrd 2764	. . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → 𝑘 ∈ ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1))))
128		elun 4112	. . . . . . . . . . . . . 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 864	. . . . . . . . . . . 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 6674	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐵)
133		ssun2 4138	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘(𝑁 + 1)) ⊆ ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1)))
134	133, 119	sseqtrrid 3987	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ ℕ0)
135	19	fdmd 6680	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐵 = ℕ0)
136	134, 135	sseqtrrd 3981	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐵)
137		funfvima2 7187	. . . . . . . . . . . . 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 2830	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (𝐵‘𝑘) ∈ {0})
144		elsni 4602	. . . . . . . . 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 7384	. . . . . . 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 11348	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑘)) = 0)
149	146, 148	eqtrd 2764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = 0)
150	106, 108, 149, 10	fsumss 15667	. . . . 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 7387	. . . 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 2774	. . 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 5195	. 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 2767	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ⊆ wss 3911  ifcif 4484  {csn 4585   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5631   “ cima 5634  Fun wfun 6493  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   + caddc 11047   · cmul 11049   ≤ cle 11185   − cmin 11381  ℕ₀cn0 12418  ℤcz 12505  ℤ_≥cuz 12769  ...cfz 13444  ↑cexp 14002  Σcsu 15628  Polycply 26122

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-sum 15629

This theorem is referenced by:  plyaddlem  26153  coeaddlem  26187