Theorem coeaddlem 24846

Description: Lemma for coeadd 24848 and dgradd 24864. (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 24817	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
2		coeadd.4	. . . . . 6 ⊢ 𝑁 = (deg‘𝐺)
3		dgrcl 24830	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
4	2, 3	eqeltrid 2894	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
5	4	adantl 485	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6		coeadd.3	. . . . . 6 ⊢ 𝑀 = (deg‘𝐹)
7		dgrcl 24830	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
8	6, 7	eqeltrid 2894	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
9	8	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
10	5, 9	ifcld 4470	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
11		addcl 10608	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
12	11	adantl 485	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13		coefv0.1	. . . . . 6 ⊢ 𝐴 = (coeff‘𝐹)
14	13	coef3 24829	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
15	14	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16		coeadd.2	. . . . . 6 ⊢ 𝐵 = (coeff‘𝐺)
17	16	coef3 24829	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
18	17	adantl 485	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19		nn0ex 11891	. . . . 5 ⊢ ℕ0 ∈ V
20	19	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21		inidm 4145	. . . 4 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
22	12, 15, 18, 20, 20, 21	off 7404	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘f + 𝐵):ℕ0⟶ℂ)
23		oveq12 7144	. . . . . . . . . 10 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = (0 + 0))
24		00id 10804	. . . . . . . . . 10 ⊢ (0 + 0) = 0
25	23, 24	eqtrdi 2849	. . . . . . . . 9 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0)
26	15	ffnd 6488	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
27	18	ffnd 6488	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
28		eqidd 2799	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
29		eqidd 2799	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
30	26, 27, 20, 20, 21, 28, 29	ofval 7398	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
31	30	eqeq1d 2800	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) = 0 ↔ ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0))
32	25, 31	syl5ibr 249	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴 ∘f + 𝐵)‘𝑘) = 0))
33	32	necon3ad 3000	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0)))
34		neorian 3081	. . . . . . 7 ⊢ (((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0) ↔ ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0))
35	33, 34	syl6ibr 255	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → ((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0)))
36	13, 6	dgrub2 24832	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
37	36	adantr 484	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
38		plyco0 24789	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
39	9, 15, 38	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
40	37, 39	mpbid 235	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
41	40	r19.21bi 3173	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
42	9	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
43	42	nn0red 11944	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
44	5	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
45	44	nn0red 11944	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
46		max1 12566	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
47	43, 45, 46	syl2anc 587	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
48		nn0re 11894	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
49	48	adantl 485	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
50	10	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
51	50	nn0red 11944	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ)
52		letr 10723	. . . . . . . . . 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)))
53	49, 43, 51, 52	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ≤ 𝑀 ∧ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
54	47, 53	mpan2d 693	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ 𝑀 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
55	41, 54	syld 47	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
56	16, 2	dgrub2 24832	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
57	56	adantl 485	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
58		plyco0 24789	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
59	5, 18, 58	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
60	57, 59	mpbid 235	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
61	60	r19.21bi 3173	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
62		max2 12568	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
63	43, 45, 62	syl2anc 587	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
64		letr 10723	. . . . . . . . . 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)))
65	49, 45, 51, 64	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ≤ 𝑁 ∧ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
66	63, 65	mpan2d 693	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ 𝑁 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
67	61, 66	syld 47	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
68	55, 67	jaod 856	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0) → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
69	35, 68	syld 47	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
70	69	ralrimiva 3149	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
71		plyco0 24789	. . . . 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))))
72	10, 22, 71	syl2anc 587	. . . 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))))
73	70, 72	mpbird 260	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 ∘f + 𝐵) “ (ℤ≥‘(if(𝑀 ≤ 𝑁, 𝑁, 𝑀) + 1))) = {0})
74		simpl 486	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
75		simpr 488	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
76	13, 6	coeid 24835	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
77	76	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
78	16, 2	coeid 24835	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
79	78	adantl 485	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
80	74, 75, 9, 5, 15, 18, 37, 57, 77, 79	plyaddlem1 24810	. . 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))))
81	1, 10, 22, 73, 80	coeeq 24824	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘f + 𝐺)) = (𝐴 ∘f + 𝐵))
82		elfznn0 12995	. . . 4 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
83		ffvelrn 6826	. . . 4 ⊢ (((𝐴 ∘f + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f + 𝐵)‘𝑘) ∈ ℂ)
84	22, 82, 83	syl2an 598	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) → ((𝐴 ∘f + 𝐵)‘𝑘) ∈ ℂ)
85	1, 10, 84, 80	dgrle 24840	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
86	81, 85	jca 515	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 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441  ifcif 4425  {csn 4525   class class class wbr 5030   ↦ cmpt 5110   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387  ℂcc 10524  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531   ≤ cle 10665  ℕ₀cn0 11885  ℤ_≥cuz 12231  ...cfz 12885  ↑cexp 13425  Σcsu 15034  Polycply 24781  coeffccoe 24783  degcdgr 24784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-0p 24274  df-ply 24785  df-coe 24787  df-dgr 24788

This theorem is referenced by:  coeadd  24848  dgradd  24864