Theorem coeeulem 25290

Description: Lemma for coeeu 25291. (Contributed by Mario Carneiro, 22-Jul-2014.)

Hypotheses

Ref	Expression
coeeu.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
coeeu.2	⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))
coeeu.3	⊢ (𝜑 → 𝐵 ∈ (ℂ ↑m ℕ0))
coeeu.4	⊢ (𝜑 → 𝑀 ∈ ℕ0)
coeeu.5	⊢ (𝜑 → 𝑁 ∈ ℕ0)
coeeu.6	⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
coeeu.7	⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
coeeu.8	⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
coeeu.9	⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))

Assertion

Ref	Expression
coeeulem	⊢ (𝜑 → 𝐴 = 𝐵)

Distinct variable groups:   𝑧,𝑘,𝐵   𝜑,𝑘,𝑧   𝐴,𝑘,𝑧   𝑘,𝑀,𝑧   𝑘,𝑁,𝑧

Allowed substitution hints:   𝑆(𝑧,𝑘)   𝐹(𝑧,𝑘)

Proof of Theorem coeeulem

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssidd 3940	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		coeeu.4	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
3		coeeu.5	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
4	2, 3	nn0addcld 12227	. . 3 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0)
5		subcl 11150	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) ∈ ℂ)
6	5	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 − 𝑦) ∈ ℂ)
7		coeeu.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))
8		cnex 10883	. . . . . . . 8 ⊢ ℂ ∈ V
9		nn0ex 12169	. . . . . . . 8 ⊢ ℕ0 ∈ V
10	8, 9	elmap 8617	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ↑m ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
11	7, 10	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
12		coeeu.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (ℂ ↑m ℕ0))
13	8, 9	elmap 8617	. . . . . . 7 ⊢ (𝐵 ∈ (ℂ ↑m ℕ0) ↔ 𝐵:ℕ0⟶ℂ)
14	12, 13	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)
15	9	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V)
16		inidm 4149	. . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
17	6, 11, 14, 15, 15, 16	off 7529	. . . . 5 ⊢ (𝜑 → (𝐴 ∘f − 𝐵):ℕ0⟶ℂ)
18	8, 9	elmap 8617	. . . . 5 ⊢ ((𝐴 ∘f − 𝐵) ∈ (ℂ ↑m ℕ0) ↔ (𝐴 ∘f − 𝐵):ℕ0⟶ℂ)
19	17, 18	sylibr 233	. . . 4 ⊢ (𝜑 → (𝐴 ∘f − 𝐵) ∈ (ℂ ↑m ℕ0))
20		0cn 10898	. . . . . . 7 ⊢ 0 ∈ ℂ
21		snssi 4738	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
22	20, 21	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
23		ssequn2 4113	. . . . . 6 ⊢ ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
24	22, 23	mpbi 229	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
25	24	oveq1i 7265	. . . 4 ⊢ ((ℂ ∪ {0}) ↑m ℕ0) = (ℂ ↑m ℕ0)
26	19, 25	eleqtrrdi 2850	. . 3 ⊢ (𝜑 → (𝐴 ∘f − 𝐵) ∈ ((ℂ ∪ {0}) ↑m ℕ0))
27	4	nn0red 12224	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℝ)
28		nn0re 12172	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
29		ltnle 10985	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝑀 + 𝑁) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑀 + 𝑁)))
30	27, 28, 29	syl2an 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑀 + 𝑁) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑀 + 𝑁)))
31	11	ffnd 6585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 Fn ℕ0)
32	14	ffnd 6585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 Fn ℕ0)
33		eqidd 2739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
34		eqidd 2739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
35	31, 32, 15, 15, 16, 33, 34	ofval 7522	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
36	35	adantrr 713	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴 ∘f − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
37	2	nn0red 12224	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
38	37	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 ∈ ℝ)
39	27	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 + 𝑁) ∈ ℝ)
40	28	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
41	40	adantrr 713	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑘 ∈ ℝ)
42	2	nn0cnd 12225	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℂ)
43	3	nn0cnd 12225	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
44	42, 43	addcomd 11107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + 𝑁) = (𝑁 + 𝑀))
45		nn0uz 12549	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 = (ℤ≥‘0)
46	3, 45	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
47	2	nn0zd 12353	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
48		eluzadd 12542	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
49	46, 47, 48	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
50	44, 49	eqeltrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑀)))
51	42	addid2d 11106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
52	51	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑀)) = (ℤ≥‘𝑀))
53	50, 52	eleqtrd 2841	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑀))
54		eluzle 12524	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑀) → 𝑀 ≤ (𝑀 + 𝑁))
55	53, 54	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ≤ (𝑀 + 𝑁))
56	55	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 ≤ (𝑀 + 𝑁))
57		simprr 769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 + 𝑁) < 𝑘)
58	38, 39, 41, 56, 57	lelttrd 11063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 < 𝑘)
59	38, 41	ltnled 11052	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑀))
60	58, 59	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ¬ 𝑘 ≤ 𝑀)
61		coeeu.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
62		plyco0 25258	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
63	2, 11, 62	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
64	61, 63	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
65	64	r19.21bi 3132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
66	65	adantrr 713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
67	66	necon1bd 2960	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
68	60, 67	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝐴‘𝑘) = 0)
69	3	nn0red 12224	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
70	69	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 ∈ ℝ)
71	2, 45	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
72	3	nn0zd 12353	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
73		eluzadd 12542	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
74	71, 72, 73	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
75	43	addid2d 11106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
76	75	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑁)) = (ℤ≥‘𝑁))
77	74, 76	eleqtrd 2841	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑁))
78		eluzle 12524	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑁) → 𝑁 ≤ (𝑀 + 𝑁))
79	77, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ≤ (𝑀 + 𝑁))
80	79	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 ≤ (𝑀 + 𝑁))
81	70, 39, 41, 80, 57	lelttrd 11063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 < 𝑘)
82	70, 41	ltnled 11052	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑁 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑁))
83	81, 82	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ¬ 𝑘 ≤ 𝑁)
84		coeeu.7	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
85		plyco0 25258	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
86	3, 14, 85	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
87	84, 86	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
88	87	r19.21bi 3132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
89	88	adantrr 713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
90	89	necon1bd 2960	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (¬ 𝑘 ≤ 𝑁 → (𝐵‘𝑘) = 0))
91	83, 90	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝐵‘𝑘) = 0)
92	68, 91	oveq12d 7273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) − (𝐵‘𝑘)) = (0 − 0))
93		0m0e0 12023	. . . . . . . . . 10 ⊢ (0 − 0) = 0
94	92, 93	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) − (𝐵‘𝑘)) = 0)
95	36, 94	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴 ∘f − 𝐵)‘𝑘) = 0)
96	95	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑀 + 𝑁) < 𝑘 → ((𝐴 ∘f − 𝐵)‘𝑘) = 0))
97	30, 96	sylbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ≤ (𝑀 + 𝑁) → ((𝐴 ∘f − 𝐵)‘𝑘) = 0))
98	97	necon1ad 2959	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁)))
99	98	ralrimiva 3107	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (((𝐴 ∘f − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁)))
100		plyco0 25258	. . . . 5 ⊢ (((𝑀 + 𝑁) ∈ ℕ0 ∧ (𝐴 ∘f − 𝐵):ℕ0⟶ℂ) → (((𝐴 ∘f − 𝐵) “ (ℤ≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴 ∘f − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁))))
101	4, 17, 100	syl2anc 583	. . . 4 ⊢ (𝜑 → (((𝐴 ∘f − 𝐵) “ (ℤ≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴 ∘f − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁))))
102	99, 101	mpbird 256	. . 3 ⊢ (𝜑 → ((𝐴 ∘f − 𝐵) “ (ℤ≥‘((𝑀 + 𝑁) + 1))) = {0})
103		df-0p 24739	. . . . 5 ⊢ 0𝑝 = (ℂ × {0})
104		fconstmpt 5640	. . . . 5 ⊢ (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
105	103, 104	eqtri 2766	. . . 4 ⊢ 0𝑝 = (𝑧 ∈ ℂ ↦ 0)
106		elfznn0 13278	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
107	35	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
108	107	oveq1d 7270	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) − (𝐵‘𝑘)) · (𝑧↑𝑘)))
109	11	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ0⟶ℂ)
110	109	ffvelrnda 6943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
111	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐵:ℕ0⟶ℂ)
112	111	ffvelrnda 6943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) ∈ ℂ)
113		expcl 13728	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
114	113	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
115	110, 112, 114	subdird 11362	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) − (𝐵‘𝑘)) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
116	108, 115	eqtrd 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
117	106, 116	sylan2 592	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → (((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
118	117	sumeq2dv 15343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
119		fzfid 13621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...(𝑀 + 𝑁)) ∈ Fin)
120	110, 114	mulcld 10926	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
121	106, 120	sylan2 592	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
122	112, 114	mulcld 10926	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
123	106, 122	sylan2 592	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
124	119, 121, 123	fsumsub 15428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))) = (Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) − Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘))))
125	119, 121	fsumcl 15373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
126		coeeu.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
127		coeeu.9	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
128	126, 127	eqtr3d 2780	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
129	128	fveq1d 6758	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧))
130	129	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧))
131		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
132		sumex 15327	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V
133		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))
134	133	fvmpt2 6868	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))
135	131, 132, 134	sylancl 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))
136		fzss2 13225	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
137	53, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
138	137	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
139	138	sselda 3917	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
140	139, 121	syldan 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
141		eldifn 4058	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → ¬ 𝑘 ∈ (0...𝑀))
142	141	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ¬ 𝑘 ∈ (0...𝑀))
143		eldifi 4057	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
144	143, 106	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ ℕ0)
145		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
146	145, 45	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
147	47	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℤ)
148		elfz5 13177	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (0...𝑀) ↔ 𝑘 ≤ 𝑀))
149	146, 147, 148	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (0...𝑀) ↔ 𝑘 ≤ 𝑀))
150	65, 149	sylibrd 258	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑀)))
151	150	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑀)))
152	151	necon1bd 2960	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ (0...𝑀) → (𝐴‘𝑘) = 0))
153	144, 152	sylan2 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (¬ 𝑘 ∈ (0...𝑀) → (𝐴‘𝑘) = 0))
154	142, 153	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) = 0)
155	154	oveq1d 7270	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
156	131, 144, 113	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
157	156	mul02d 11103	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
158	155, 157	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
159	138, 140, 158, 119	fsumss 15365	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)))
160	135, 159	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)))
161		sumex 15327	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V
162		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
163	162	fvmpt2 6868	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
164	131, 161, 163	sylancl 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
165		fzss2 13225	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑁) → (0...𝑁) ⊆ (0...(𝑀 + 𝑁)))
166	77, 165	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝑁) ⊆ (0...(𝑀 + 𝑁)))
167	166	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ⊆ (0...(𝑀 + 𝑁)))
168	167	sselda 3917	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
169	168, 123	syldan 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
170		eldifn 4058	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁)) → ¬ 𝑘 ∈ (0...𝑁))
171	170	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ¬ 𝑘 ∈ (0...𝑁))
172		eldifi 4057	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
173	172, 106	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁)) → 𝑘 ∈ ℕ0)
174	72	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℤ)
175		elfz5 13177	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ≤ 𝑁))
176	146, 174, 175	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ≤ 𝑁))
177	88, 176	sylibrd 258	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑁)))
178	177	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑁)))
179	178	necon1bd 2960	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ (0...𝑁) → (𝐵‘𝑘) = 0))
180	173, 179	sylan2 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (¬ 𝑘 ∈ (0...𝑁) → (𝐵‘𝑘) = 0))
181	171, 180	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (𝐵‘𝑘) = 0)
182	181	oveq1d 7270	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
183	131, 173, 113	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (𝑧↑𝑘) ∈ ℂ)
184	183	mul02d 11103	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑘)) = 0)
185	182, 184	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = 0)
186	167, 169, 185, 119	fsumss 15365	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
187	164, 186	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
188	130, 160, 187	3eqtr3d 2786	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
189	125, 188	subeq0bd 11331	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) − Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘))) = 0)
190	118, 124, 189	3eqtrrd 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 0 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)))
191	190	mpteq2dva 5170	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ 0) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘))))
192	105, 191	syl5eq 2791	. . 3 ⊢ (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘))))
193	1, 4, 26, 102, 192	plyeq0 25277	. 2 ⊢ (𝜑 → (𝐴 ∘f − 𝐵) = (ℕ0 × {0}))
194		ofsubeq0 11900	. . 3 ⊢ ((ℕ0 ∈ V ∧ 𝐴:ℕ0⟶ℂ ∧ 𝐵:ℕ0⟶ℂ) → ((𝐴 ∘f − 𝐵) = (ℕ0 × {0}) ↔ 𝐴 = 𝐵))
195	9, 11, 14, 194	mp3an2i 1464	. 2 ⊢ (𝜑 → ((𝐴 ∘f − 𝐵) = (ℕ0 × {0}) ↔ 𝐴 = 𝐵))
196	193, 195	mpbid 231	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578   “ cima 5583  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509   ↑_m cmap 8573  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807   < clt 10940   ≤ cle 10941   − cmin 11135  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  ↑cexp 13710  Σcsu 15325  0_𝑝c0p 24738  Polycply 25250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-0p 24739

This theorem is referenced by:  coeeu  25291