Theorem coeaddlem 24005

Description: Lemma for coeadd 24007 and dgradd 24023. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
coefv0.1	⊢ 𝐴 = (coeff‘𝐹)
coeadd.2	⊢ 𝐵 = (coeff‘𝐺)
coeadd.3	⊢ 𝑀 = (deg‘𝐹)
coeadd.4	⊢ 𝑁 = (deg‘𝐺)

Assertion

Ref	Expression
coeaddlem	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘𝑓 + 𝐺)) = (𝐴 ∘𝑓 + 𝐵) ∧ (deg‘(𝐹 ∘𝑓 + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))

Proof of Theorem coeaddlem

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

Step	Hyp	Ref	Expression
1		plyaddcl 23976	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘ℂ))
2		coeadd.4	. . . . . 6 ⊢ 𝑁 = (deg‘𝐺)
3		dgrcl 23989	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
4	2, 3	syl5eqel 2705	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
5	4	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6		coeadd.3	. . . . . 6 ⊢ 𝑀 = (deg‘𝐹)
7		dgrcl 23989	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
8	6, 7	syl5eqel 2705	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
9	8	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
10	5, 9	ifcld 4131	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
11		addcl 10018	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
12	11	adantl 482	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13		coefv0.1	. . . . . 6 ⊢ 𝐴 = (coeff‘𝐹)
14	13	coef3 23988	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
15	14	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16		coeadd.2	. . . . . 6 ⊢ 𝐵 = (coeff‘𝐺)
17	16	coef3 23988	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
18	17	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19		nn0ex 11298	. . . . 5 ⊢ ℕ0 ∈ V
20	19	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21		inidm 3822	. . . 4 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
22	12, 15, 18, 20, 20, 21	off 6912	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘𝑓 + 𝐵):ℕ0⟶ℂ)
23		oveq12 6659	. . . . . . . . . 10 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = (0 + 0))
24		00id 10211	. . . . . . . . . 10 ⊢ (0 + 0) = 0
25	23, 24	syl6eq 2672	. . . . . . . . 9 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0)
26		ffn 6045	. . . . . . . . . . . 12 ⊢ (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
27	15, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
28		ffn 6045	. . . . . . . . . . . 12 ⊢ (𝐵:ℕ0⟶ℂ → 𝐵 Fn ℕ0)
29	18, 28	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
30		eqidd 2623	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
31		eqidd 2623	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
32	27, 29, 20, 20, 21, 30, 31	ofval 6906	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
33	32	eqeq1d 2624	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘𝑓 + 𝐵)‘𝑘) = 0 ↔ ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0))
34	25, 33	syl5ibr 236	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) = 0))
35	34	necon3ad 2807	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘𝑓 + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0)))
36		neorian 2888	. . . . . . 7 ⊢ (((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0) ↔ ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0))
37	35, 36	syl6ibr 242	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘𝑓 + 𝐵)‘𝑘) ≠ 0 → ((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0)))
38	13, 6	dgrub2 23991	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
39	38	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
40		plyco0 23948	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
41	9, 15, 40	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
42	39, 41	mpbid 222	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
43	42	r19.21bi 2932	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
44	9	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
45	44	nn0red 11352	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
46	5	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
47	46	nn0red 11352	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
48		max1 12016	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
49	45, 47, 48	syl2anc 693	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
50		nn0re 11301	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
51	50	adantl 482	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
52	10	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
53	52	nn0red 11352	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ)
54		letr 10131	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘 ≤ 𝑀 ∧ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
55	51, 45, 53, 54	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ≤ 𝑀 ∧ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
56	49, 55	mpan2d 710	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ 𝑀 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
57	43, 56	syld 47	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
58	16, 2	dgrub2 23991	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
59	58	adantl 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
60		plyco0 23948	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
61	5, 18, 60	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
62	59, 61	mpbid 222	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
63	62	r19.21bi 2932	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
64		max2 12018	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
65	45, 47, 64	syl2anc 693	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
66		letr 10131	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘 ≤ 𝑁 ∧ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
67	51, 47, 53, 66	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ≤ 𝑁 ∧ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
68	65, 67	mpan2d 710	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ 𝑁 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
69	63, 68	syld 47	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
70	57, 69	jaod 395	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
71	37, 70	syld 47	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
72	71	ralrimiva 2966	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴 ∘𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
73		plyco0 23948	. . . . 5 ⊢ ((if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0 ∧ (𝐴 ∘𝑓 + 𝐵):ℕ0⟶ℂ) → (((𝐴 ∘𝑓 + 𝐵) “ (ℤ≥‘(if(𝑀 ≤ 𝑁, 𝑁, 𝑀) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴 ∘𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
74	10, 22, 73	syl2anc 693	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((𝐴 ∘𝑓 + 𝐵) “ (ℤ≥‘(if(𝑀 ≤ 𝑁, 𝑁, 𝑀) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴 ∘𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
75	72, 74	mpbird 247	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 ∘𝑓 + 𝐵) “ (ℤ≥‘(if(𝑀 ≤ 𝑁, 𝑁, 𝑀) + 1))) = {0})
76		simpl 473	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
77		simpr 477	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
78	13, 6	coeid 23994	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
79	78	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
80	16, 2	coeid 23994	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
81	80	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
82	76, 77, 9, 5, 15, 18, 39, 59, 79, 81	plyaddlem1 23969	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))))
83	1, 10, 22, 75, 82	coeeq 23983	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 + 𝐺)) = (𝐴 ∘𝑓 + 𝐵))
84		elfznn0 12433	. . . 4 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
85		ffvelrn 6357	. . . 4 ⊢ (((𝐴 ∘𝑓 + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ ℂ)
86	22, 84, 85	syl2an 494	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴 ∘𝑓 + 𝐵)‘𝑘) ∈ ℂ)
87	1, 10, 86, 82	dgrle 23999	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘𝑓 + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
88	83, 87	jca 554	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘𝑓 + 𝐺)) = (𝐴 ∘𝑓 + 𝐵) ∧ (deg‘(𝐹 ∘𝑓 + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200  ifcif 4086  {csn 4177   class class class wbr 4653   ↦ cmpt 4729   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   ≤ cle 10075  ℕ₀cn0 11292  ℤ_≥cuz 11687  ...cfz 12326  ↑cexp 12860  Σcsu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  coeadd  24007  dgradd  24023