Theorem coeeulem 25118

Description: Lemma for coeeu 25119. (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 3924	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		coeeu.4	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
3		coeeu.5	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
4	2, 3	nn0addcld 12154	. . 3 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0)
5		subcl 11077	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) ∈ ℂ)
6	5	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 − 𝑦) ∈ ℂ)
7		coeeu.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))
8		cnex 10810	. . . . . . . 8 ⊢ ℂ ∈ V
9		nn0ex 12096	. . . . . . . 8 ⊢ ℕ0 ∈ V
10	8, 9	elmap 8552	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ↑m ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
11	7, 10	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
12		coeeu.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (ℂ ↑m ℕ0))
13	8, 9	elmap 8552	. . . . . . 7 ⊢ (𝐵 ∈ (ℂ ↑m ℕ0) ↔ 𝐵:ℕ0⟶ℂ)
14	12, 13	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)
15	9	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V)
16		inidm 4133	. . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
17	6, 11, 14, 15, 15, 16	off 7486	. . . . 5 ⊢ (𝜑 → (𝐴 ∘f − 𝐵):ℕ0⟶ℂ)
18	8, 9	elmap 8552	. . . . 5 ⊢ ((𝐴 ∘f − 𝐵) ∈ (ℂ ↑m ℕ0) ↔ (𝐴 ∘f − 𝐵):ℕ0⟶ℂ)
19	17, 18	sylibr 237	. . . 4 ⊢ (𝜑 → (𝐴 ∘f − 𝐵) ∈ (ℂ ↑m ℕ0))
20		0cn 10825	. . . . . . 7 ⊢ 0 ∈ ℂ
21		snssi 4721	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
22	20, 21	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
23		ssequn2 4097	. . . . . 6 ⊢ ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
24	22, 23	mpbi 233	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
25	24	oveq1i 7223	. . . 4 ⊢ ((ℂ ∪ {0}) ↑m ℕ0) = (ℂ ↑m ℕ0)
26	19, 25	eleqtrrdi 2849	. . 3 ⊢ (𝜑 → (𝐴 ∘f − 𝐵) ∈ ((ℂ ∪ {0}) ↑m ℕ0))
27	4	nn0red 12151	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℝ)
28		nn0re 12099	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
29		ltnle 10912	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝑀 + 𝑁) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑀 + 𝑁)))
30	27, 28, 29	syl2an 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑀 + 𝑁) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑀 + 𝑁)))
31	11	ffnd 6546	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 Fn ℕ0)
32	14	ffnd 6546	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 Fn ℕ0)
33		eqidd 2738	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
34		eqidd 2738	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
35	31, 32, 15, 15, 16, 33, 34	ofval 7479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
36	35	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴 ∘f − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
37	2	nn0red 12151	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
38	37	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 ∈ ℝ)
39	27	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 + 𝑁) ∈ ℝ)
40	28	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
41	40	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑘 ∈ ℝ)
42	2	nn0cnd 12152	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℂ)
43	3	nn0cnd 12152	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
44	42, 43	addcomd 11034	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + 𝑁) = (𝑁 + 𝑀))
45		nn0uz 12476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 = (ℤ≥‘0)
46	3, 45	eleqtrdi 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
47	2	nn0zd 12280	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
48		eluzadd 12469	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
49	46, 47, 48	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
50	44, 49	eqeltrd 2838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑀)))
51	42	addid2d 11033	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
52	51	fveq2d 6721	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑀)) = (ℤ≥‘𝑀))
53	50, 52	eleqtrd 2840	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑀))
54		eluzle 12451	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑀) → 𝑀 ≤ (𝑀 + 𝑁))
55	53, 54	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ≤ (𝑀 + 𝑁))
56	55	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 ≤ (𝑀 + 𝑁))
57		simprr 773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 + 𝑁) < 𝑘)
58	38, 39, 41, 56, 57	lelttrd 10990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 < 𝑘)
59	38, 41	ltnled 10979	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑀))
60	58, 59	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ¬ 𝑘 ≤ 𝑀)
61		coeeu.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
62		plyco0 25086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
63	2, 11, 62	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
64	61, 63	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
65	64	r19.21bi 3130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
66	65	adantrr 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
67	66	necon1bd 2958	. . . . . . . . . . . 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 12151	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
70	69	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 ∈ ℝ)
71	2, 45	eleqtrdi 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
72	3	nn0zd 12280	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
73		eluzadd 12469	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
74	71, 72, 73	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
75	43	addid2d 11033	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
76	75	fveq2d 6721	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑁)) = (ℤ≥‘𝑁))
77	74, 76	eleqtrd 2840	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑁))
78		eluzle 12451	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑁) → 𝑁 ≤ (𝑀 + 𝑁))
79	77, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ≤ (𝑀 + 𝑁))
80	79	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 ≤ (𝑀 + 𝑁))
81	70, 39, 41, 80, 57	lelttrd 10990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 < 𝑘)
82	70, 41	ltnled 10979	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑁 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑁))
83	81, 82	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ¬ 𝑘 ≤ 𝑁)
84		coeeu.7	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
85		plyco0 25086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
86	3, 14, 85	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
87	84, 86	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
88	87	r19.21bi 3130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
89	88	adantrr 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
90	89	necon1bd 2958	. . . . . . . . . . . 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 7231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) − (𝐵‘𝑘)) = (0 − 0))
93		0m0e0 11950	. . . . . . . . . 10 ⊢ (0 − 0) = 0
94	92, 93	eqtrdi 2794	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) − (𝐵‘𝑘)) = 0)
95	36, 94	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴 ∘f − 𝐵)‘𝑘) = 0)
96	95	expr 460	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑀 + 𝑁) < 𝑘 → ((𝐴 ∘f − 𝐵)‘𝑘) = 0))
97	30, 96	sylbird 263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ≤ (𝑀 + 𝑁) → ((𝐴 ∘f − 𝐵)‘𝑘) = 0))
98	97	necon1ad 2957	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁)))
99	98	ralrimiva 3105	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (((𝐴 ∘f − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁)))
100		plyco0 25086	. . . . 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 587	. . . 4 ⊢ (𝜑 → (((𝐴 ∘f − 𝐵) “ (ℤ≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴 ∘f − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁))))
102	99, 101	mpbird 260	. . 3 ⊢ (𝜑 → ((𝐴 ∘f − 𝐵) “ (ℤ≥‘((𝑀 + 𝑁) + 1))) = {0})
103		df-0p 24567	. . . . 5 ⊢ 0𝑝 = (ℂ × {0})
104		fconstmpt 5611	. . . . 5 ⊢ (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
105	103, 104	eqtri 2765	. . . 4 ⊢ 0𝑝 = (𝑧 ∈ ℂ ↦ 0)
106		elfznn0 13205	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
107	35	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
108	107	oveq1d 7228	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) − (𝐵‘𝑘)) · (𝑧↑𝑘)))
109	11	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ0⟶ℂ)
110	109	ffvelrnda 6904	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
111	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐵:ℕ0⟶ℂ)
112	111	ffvelrnda 6904	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) ∈ ℂ)
113		expcl 13653	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
114	113	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
115	110, 112, 114	subdird 11289	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) − (𝐵‘𝑘)) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
116	108, 115	eqtrd 2777	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
117	106, 116	sylan2 596	. . . . . . 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 15267	. . . . . 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 13546	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...(𝑀 + 𝑁)) ∈ Fin)
120	110, 114	mulcld 10853	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
121	106, 120	sylan2 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
122	112, 114	mulcld 10853	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
123	106, 122	sylan2 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
124	119, 121, 123	fsumsub 15352	. . . . . 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 15297	. . . . . . 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 2779	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
129	128	fveq1d 6719	. . . . . . . . 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 484	. . . . . . . 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 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
132		sumex 15251	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V
133		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))
134	133	fvmpt2 6829	. . . . . . . . . 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 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))
136		fzss2 13152	. . . . . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
139	138	sselda 3901	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
140	139, 121	syldan 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
141		eldifn 4042	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → ¬ 𝑘 ∈ (0...𝑀))
142	141	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ¬ 𝑘 ∈ (0...𝑀))
143		eldifi 4041	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
146	145, 45	eleqtrdi 2848	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
147	47	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℤ)
148		elfz5 13104	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (0...𝑀) ↔ 𝑘 ≤ 𝑀))
149	146, 147, 148	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (0...𝑀) ↔ 𝑘 ≤ 𝑀))
150	65, 149	sylibrd 262	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑀)))
151	150	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑀)))
152	151	necon1bd 2958	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ (0...𝑀) → (𝐴‘𝑘) = 0))
153	144, 152	sylan2 596	. . . . . . . . . . . . 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 7228	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
156	131, 144, 113	syl2an 599	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
157	156	mul02d 11030	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
158	155, 157	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
159	138, 140, 158, 119	fsumss 15289	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)))
160	135, 159	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)))
161		sumex 15251	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V
162		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
163	162	fvmpt2 6829	. . . . . . . . . 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 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
165		fzss2 13152	. . . . . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ⊆ (0...(𝑀 + 𝑁)))
168	167	sselda 3901	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
169	168, 123	syldan 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
170		eldifn 4042	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁)) → ¬ 𝑘 ∈ (0...𝑁))
171	170	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ¬ 𝑘 ∈ (0...𝑁))
172		eldifi 4041	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℤ)
175		elfz5 13104	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ≤ 𝑁))
176	146, 174, 175	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ≤ 𝑁))
177	88, 176	sylibrd 262	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑁)))
178	177	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑁)))
179	178	necon1bd 2958	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ (0...𝑁) → (𝐵‘𝑘) = 0))
180	173, 179	sylan2 596	. . . . . . . . . . . . 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 7228	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
183	131, 173, 113	syl2an 599	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (𝑧↑𝑘) ∈ ℂ)
184	183	mul02d 11030	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑘)) = 0)
185	182, 184	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = 0)
186	167, 169, 185, 119	fsumss 15289	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
187	164, 186	eqtrd 2777	. . . . . . . 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 2785	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
189	125, 188	subeq0bd 11258	. . . . . 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 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 0 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘)))
191	190	mpteq2dva 5150	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ 0) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘))))
192	105, 191	syl5eq 2790	. . 3 ⊢ (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘f − 𝐵)‘𝑘) · (𝑧↑𝑘))))
193	1, 4, 26, 102, 192	plyeq0 25105	. 2 ⊢ (𝜑 → (𝐴 ∘f − 𝐵) = (ℕ0 × {0}))
194		ofsubeq0 11827	. . 3 ⊢ ((ℕ0 ∈ V ∧ 𝐴:ℕ0⟶ℂ ∧ 𝐵:ℕ0⟶ℂ) → ((𝐴 ∘f − 𝐵) = (ℕ0 × {0}) ↔ 𝐴 = 𝐵))
195	9, 11, 14, 194	mp3an2i 1468	. 2 ⊢ (𝜑 → ((𝐴 ∘f − 𝐵) = (ℕ0 × {0}) ↔ 𝐴 = 𝐵))
196	193, 195	mpbid 235	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  Vcvv 3408   ∖ cdif 3863   ∪ cun 3864   ⊆ wss 3866  {csn 4541   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549   “ cima 5554  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ∘_f cof 7467   ↑_m cmap 8508  ℂcc 10727  ℝcr 10728  0cc0 10729  1c1 10730   + caddc 10732   · cmul 10734   < clt 10867   ≤ cle 10868   − cmin 11062  ℕ₀cn0 12090  ℤcz 12176  ℤ_≥cuz 12438  ...cfz 13095  ↑cexp 13635  Σcsu 15249  0_𝑝c0p 24566  Polycply 25078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-inf 9059  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-rlim 15050  df-sum 15250  df-0p 24567

This theorem is referenced by:  coeeu  25119