Theorem coeeulem 23784

Description: Lemma for coeeu 23785. (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		ssid 3587	. . . 4 ⊢ ℂ ⊆ ℂ
2	1	a1i 11	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
3		coeeu.4	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
4		coeeu.5	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
5	3, 4	nn0addcld 11232	. . 3 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0)
6		subcl 10159	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) ∈ ℂ)
7	6	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 − 𝑦) ∈ ℂ)
8		coeeu.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑𝑚 ℕ0))
9		cnex 9896	. . . . . . . 8 ⊢ ℂ ∈ V
10		nn0ex 11175	. . . . . . . 8 ⊢ ℕ0 ∈ V
11	9, 10	elmap 7772	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ↑𝑚 ℕ0) ↔ 𝐴:ℕ0⟶ℂ)
12	8, 11	sylib 207	. . . . . 6 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
13		coeeu.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (ℂ ↑𝑚 ℕ0))
14	9, 10	elmap 7772	. . . . . . 7 ⊢ (𝐵 ∈ (ℂ ↑𝑚 ℕ0) ↔ 𝐵:ℕ0⟶ℂ)
15	13, 14	sylib 207	. . . . . 6 ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)
16	10	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V)
17		inidm 3784	. . . . . 6 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
18	7, 12, 15, 16, 16, 17	off 6810	. . . . 5 ⊢ (𝜑 → (𝐴 ∘𝑓 − 𝐵):ℕ0⟶ℂ)
19	9, 10	elmap 7772	. . . . 5 ⊢ ((𝐴 ∘𝑓 − 𝐵) ∈ (ℂ ↑𝑚 ℕ0) ↔ (𝐴 ∘𝑓 − 𝐵):ℕ0⟶ℂ)
20	18, 19	sylibr 223	. . . 4 ⊢ (𝜑 → (𝐴 ∘𝑓 − 𝐵) ∈ (ℂ ↑𝑚 ℕ0))
21		0cn 9911	. . . . . . 7 ⊢ 0 ∈ ℂ
22		snssi 4280	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
23	21, 22	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
24		ssequn2 3748	. . . . . 6 ⊢ ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
25	23, 24	mpbi 219	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
26	25	oveq1i 6559	. . . 4 ⊢ ((ℂ ∪ {0}) ↑𝑚 ℕ0) = (ℂ ↑𝑚 ℕ0)
27	20, 26	syl6eleqr 2699	. . 3 ⊢ (𝜑 → (𝐴 ∘𝑓 − 𝐵) ∈ ((ℂ ∪ {0}) ↑𝑚 ℕ0))
28	5	nn0red 11229	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℝ)
29		nn0re 11178	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
30		ltnle 9996	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝑀 + 𝑁) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑀 + 𝑁)))
31	28, 29, 30	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑀 + 𝑁) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑀 + 𝑁)))
32		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
33	12, 32	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 Fn ℕ0)
34		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝐵:ℕ0⟶ℂ → 𝐵 Fn ℕ0)
35	15, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 Fn ℕ0)
36		eqidd 2611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
37		eqidd 2611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
38	33, 35, 16, 16, 17, 36, 37	ofval 6804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘𝑓 − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
39	38	adantrr 749	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴 ∘𝑓 − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
40	3	nn0red 11229	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℝ)
41	40	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 ∈ ℝ)
42	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 + 𝑁) ∈ ℝ)
43	29	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
44	43	adantrr 749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑘 ∈ ℝ)
45	3	nn0cnd 11230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℂ)
46	4	nn0cnd 11230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
47	45, 46	addcomd 10117	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + 𝑁) = (𝑁 + 𝑀))
48		nn0uz 11598	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ0 = (ℤ≥‘0)
49	4, 48	syl6eleq 2698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
50	3	nn0zd 11356	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
51		eluzadd 11592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
52	49, 50, 51	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ (ℤ≥‘(0 + 𝑀)))
53	47, 52	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑀)))
54	45	addid2d 10116	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
55	54	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑀)) = (ℤ≥‘𝑀))
56	53, 55	eleqtrd 2690	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑀))
57		eluzle 11576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑀) → 𝑀 ≤ (𝑀 + 𝑁))
58	56, 57	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ≤ (𝑀 + 𝑁))
59	58	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 ≤ (𝑀 + 𝑁))
60		simprr 792	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 + 𝑁) < 𝑘)
61	41, 42, 44, 59, 60	lelttrd 10074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑀 < 𝑘)
62	41, 44	ltnled 10063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑀 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑀))
63	61, 62	mpbid 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ¬ 𝑘 ≤ 𝑀)
64		coeeu.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
65		plyco0 23752	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
66	3, 12, 65	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
67	64, 66	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
68	67	r19.21bi 2916	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
69	68	adantrr 749	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
70	69	necon1bd 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
71	63, 70	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝐴‘𝑘) = 0)
72	4	nn0red 11229	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ)
73	72	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 ∈ ℝ)
74	3, 48	syl6eleq 2698	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
75	4	nn0zd 11356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76		eluzadd 11592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
77	74, 75, 76	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘(0 + 𝑁)))
78	46	addid2d 10116	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
79	78	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤ≥‘(0 + 𝑁)) = (ℤ≥‘𝑁))
80	77, 79	eleqtrd 2690	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ (ℤ≥‘𝑁))
81		eluzle 11576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑁) → 𝑁 ≤ (𝑀 + 𝑁))
82	80, 81	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ≤ (𝑀 + 𝑁))
83	82	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 ≤ (𝑀 + 𝑁))
84	73, 42, 44, 83, 60	lelttrd 10074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → 𝑁 < 𝑘)
85	73, 44	ltnled 10063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝑁 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑁))
86	84, 85	mpbid 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ¬ 𝑘 ≤ 𝑁)
87		coeeu.7	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
88		plyco0 23752	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
89	4, 15, 88	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
90	87, 89	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
91	90	r19.21bi 2916	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
92	91	adantrr 749	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
93	92	necon1bd 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (¬ 𝑘 ≤ 𝑁 → (𝐵‘𝑘) = 0))
94	86, 93	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → (𝐵‘𝑘) = 0)
95	71, 94	oveq12d 6567	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) − (𝐵‘𝑘)) = (0 − 0))
96		0m0e0 11007	. . . . . . . . . 10 ⊢ (0 − 0) = 0
97	95, 96	syl6eq 2660	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴‘𝑘) − (𝐵‘𝑘)) = 0)
98	39, 97	eqtrd 2644	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ0 ∧ (𝑀 + 𝑁) < 𝑘)) → ((𝐴 ∘𝑓 − 𝐵)‘𝑘) = 0)
99	98	expr 641	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑀 + 𝑁) < 𝑘 → ((𝐴 ∘𝑓 − 𝐵)‘𝑘) = 0))
100	31, 99	sylbird 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ≤ (𝑀 + 𝑁) → ((𝐴 ∘𝑓 − 𝐵)‘𝑘) = 0))
101	100	necon1ad 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘𝑓 − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁)))
102	101	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (((𝐴 ∘𝑓 − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁)))
103		plyco0 23752	. . . . 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))))
104	5, 18, 103	syl2anc 691	. . . 4 ⊢ (𝜑 → (((𝐴 ∘𝑓 − 𝐵) “ (ℤ≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴 ∘𝑓 − 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ (𝑀 + 𝑁))))
105	102, 104	mpbird 246	. . 3 ⊢ (𝜑 → ((𝐴 ∘𝑓 − 𝐵) “ (ℤ≥‘((𝑀 + 𝑁) + 1))) = {0})
106		df-0p 23243	. . . . 5 ⊢ 0𝑝 = (ℂ × {0})
107		fconstmpt 5085	. . . . 5 ⊢ (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
108	106, 107	eqtri 2632	. . . 4 ⊢ 0𝑝 = (𝑧 ∈ ℂ ↦ 0)
109		elfznn0 12302	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
110	38	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘𝑓 − 𝐵)‘𝑘) = ((𝐴‘𝑘) − (𝐵‘𝑘)))
111	110	oveq1d 6564	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘𝑓 − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) − (𝐵‘𝑘)) · (𝑧↑𝑘)))
112	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ0⟶ℂ)
113	112	ffvelrnda 6267	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ)
114	15	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐵:ℕ0⟶ℂ)
115	114	ffvelrnda 6267	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) ∈ ℂ)
116		expcl 12740	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
117	116	adantll 746	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
118	113, 115, 117	subdird 10366	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) − (𝐵‘𝑘)) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
119	111, 118	eqtrd 2644	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘𝑓 − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
120	109, 119	sylan2 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → (((𝐴 ∘𝑓 − 𝐵)‘𝑘) · (𝑧↑𝑘)) = (((𝐴‘𝑘) · (𝑧↑𝑘)) − ((𝐵‘𝑘) · (𝑧↑𝑘))))
121	120	sumeq2dv 14281	. . . . . 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))))
122		fzfid 12634	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...(𝑀 + 𝑁)) ∈ Fin)
123	113, 117	mulcld 9939	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
124	109, 123	sylan2 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
125	115, 117	mulcld 9939	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
126	109, 125	sylan2 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑀 + 𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
127	122, 124, 126	fsumsub 14362	. . . . . 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))))
128	122, 124	fsumcl 14311	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
129		coeeu.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
130		coeeu.9	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
131	129, 130	eqtr3d 2646	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
132	131	fveq1d 6105	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧))
133	132	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))
134		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
135		sumex 14266	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ V
136		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))
137	136	fvmpt2 6200	. . . . . . . . . 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)))
138	134, 135, 137	sylancl 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))
139		fzss2 12252	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
140	56, 139	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
141	140	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑀) ⊆ (0...(𝑀 + 𝑁)))
142	141	sselda 3568	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
143	142, 124	syldan 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑀)) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
144		eldifn 3695	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → ¬ 𝑘 ∈ (0...𝑀))
145	144	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ¬ 𝑘 ∈ (0...𝑀))
146		eldifi 3694	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
147	146, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀)) → 𝑘 ∈ ℕ0)
148		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
149	148, 48	syl6eleq 2698	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
150	50	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℤ)
151		elfz5 12205	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ≥‘0) ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (0...𝑀) ↔ 𝑘 ≤ 𝑀))
152	149, 150, 151	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (0...𝑀) ↔ 𝑘 ≤ 𝑀))
153	68, 152	sylibrd 248	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑀)))
154	153	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑀)))
155	154	necon1bd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ (0...𝑀) → (𝐴‘𝑘) = 0))
156	147, 155	sylan2 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (¬ 𝑘 ∈ (0...𝑀) → (𝐴‘𝑘) = 0))
157	145, 156	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝐴‘𝑘) = 0)
158	157	oveq1d 6564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
159	134, 147, 116	syl2an 493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (𝑧↑𝑘) ∈ ℂ)
160	159	mul02d 10113	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → (0 · (𝑧↑𝑘)) = 0)
161	158, 160	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑀))) → ((𝐴‘𝑘) · (𝑧↑𝑘)) = 0)
162	141, 143, 161, 122	fsumss 14303	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)))
163	138, 162	eqtrd 2644	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)))
164		sumex 14266	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ V
165		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
166	165	fvmpt2 6200	. . . . . . . . . 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)))
167	134, 164, 166	sylancl 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))
168		fzss2 12252	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝑁) → (0...𝑁) ⊆ (0...(𝑀 + 𝑁)))
169	80, 168	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝑁) ⊆ (0...(𝑀 + 𝑁)))
170	169	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ⊆ (0...(𝑀 + 𝑁)))
171	170	sselda 3568	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
172	171, 126	syldan 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐵‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
173		eldifn 3695	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁)) → ¬ 𝑘 ∈ (0...𝑁))
174	173	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ¬ 𝑘 ∈ (0...𝑁))
175		eldifi 3694	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁)) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
176	175, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁)) → 𝑘 ∈ ℕ0)
177	75	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℤ)
178		elfz5 12205	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ≤ 𝑁))
179	149, 177, 178	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ≤ 𝑁))
180	91, 179	sylibrd 248	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑁)))
181	180	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑁)))
182	181	necon1bd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ (0...𝑁) → (𝐵‘𝑘) = 0))
183	176, 182	sylan2 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (¬ 𝑘 ∈ (0...𝑁) → (𝐵‘𝑘) = 0))
184	174, 183	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (𝐵‘𝑘) = 0)
185	184	oveq1d 6564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
186	134, 176, 116	syl2an 493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (𝑧↑𝑘) ∈ ℂ)
187	186	mul02d 10113	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → (0 · (𝑧↑𝑘)) = 0)
188	185, 187	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ (0...𝑁))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = 0)
189	170, 172, 188, 122	fsumss 14303	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
190	167, 189	eqtrd 2644	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))‘𝑧) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
191	133, 163, 190	3eqtr3d 2652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘)))
192	128, 191	subeq0bd 10335	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝑧↑𝑘)) − Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐵‘𝑘) · (𝑧↑𝑘))) = 0)
193	121, 127, 192	3eqtrrd 2649	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 0 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘𝑓 − 𝐵)‘𝑘) · (𝑧↑𝑘)))
194	193	mpteq2dva 4672	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ 0) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘𝑓 − 𝐵)‘𝑘) · (𝑧↑𝑘))))
195	108, 194	syl5eq 2656	. . 3 ⊢ (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑀 + 𝑁))(((𝐴 ∘𝑓 − 𝐵)‘𝑘) · (𝑧↑𝑘))))
196	2, 5, 27, 105, 195	plyeq0 23771	. 2 ⊢ (𝜑 → (𝐴 ∘𝑓 − 𝐵) = (ℕ0 × {0}))
197		ofsubeq0 10894	. . 3 ⊢ ((ℕ0 ∈ V ∧ 𝐴:ℕ0⟶ℂ ∧ 𝐵:ℕ0⟶ℂ) → ((𝐴 ∘𝑓 − 𝐵) = (ℕ0 × {0}) ↔ 𝐴 = 𝐵))
198	16, 12, 15, 197	syl3anc 1318	. 2 ⊢ (𝜑 → ((𝐴 ∘𝑓 − 𝐵) = (ℕ0 × {0}) ↔ 𝐴 = 𝐵))
199	196, 198	mpbid 221	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ↑_𝑚 cmap 7744  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ↑cexp 12722  Σcsu 14264  0_𝑝c0p 23242  Polycply 23744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243

This theorem is referenced by:  coeeu  23785