Theorem coeaddlem 26214

Description: Lemma for coeadd 26216 and dgradd 26232. (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 26185	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
2		coeadd.4	. . . . . 6 ⊢ 𝑁 = (deg‘𝐺)
3		dgrcl 26198	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
4	2, 3	eqeltrid 2840	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
5	4	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6		coeadd.3	. . . . . 6 ⊢ 𝑀 = (deg‘𝐹)
7		dgrcl 26198	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
8	6, 7	eqeltrid 2840	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
9	8	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
10	5, 9	ifcld 4513	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
11		addcl 11120	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
12	11	adantl 481	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13		coefv0.1	. . . . . 6 ⊢ 𝐴 = (coeff‘𝐹)
14	13	coef3 26197	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
15	14	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16		coeadd.2	. . . . . 6 ⊢ 𝐵 = (coeff‘𝐺)
17	16	coef3 26197	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
18	17	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19		nn0ex 12443	. . . . 5 ⊢ ℕ0 ∈ V
20	19	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21		inidm 4167	. . . 4 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
22	12, 15, 18, 20, 20, 21	off 7649	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘f + 𝐵):ℕ0⟶ℂ)
23		oveq12 7376	. . . . . . . . . 10 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = (0 + 0))
24		00id 11321	. . . . . . . . . 10 ⊢ (0 + 0) = 0
25	23, 24	eqtrdi 2787	. . . . . . . . 9 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0)
26	15	ffnd 6669	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
27	18	ffnd 6669	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
28		eqidd 2737	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
29		eqidd 2737	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
30	26, 27, 20, 20, 21, 28, 29	ofval 7642	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
31	30	eqeq1d 2738	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) = 0 ↔ ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0))
32	25, 31	imbitrrid 246	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴 ∘f + 𝐵)‘𝑘) = 0))
33	32	necon3ad 2945	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0)))
34		neorian 3027	. . . . . . 7 ⊢ (((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0) ↔ ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0))
35	33, 34	imbitrrdi 252	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → ((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0)))
36	13, 6	dgrub2 26200	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
37	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
38		plyco0 26157	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
39	9, 15, 38	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
40	37, 39	mpbid 232	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
41	40	r19.21bi 3229	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
42	9	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
43	42	nn0red 12499	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
44	5	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
45	44	nn0red 12499	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
46		max1 13137	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
47	43, 45, 46	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
48		nn0re 12446	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
49	48	adantl 481	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
50	10	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
51	50	nn0red 12499	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ)
52		letr 11240	. . . . . . . . . 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 1374	. . . . . . . . 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 695	. . . . . . . 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 26200	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
57	56	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
58		plyco0 26157	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
59	5, 18, 58	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
60	57, 59	mpbid 232	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
61	60	r19.21bi 3229	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
62		max2 13139	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
63	43, 45, 62	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
64		letr 11240	. . . . . . . . . 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 1374	. . . . . . . . 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 695	. . . . . . . 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 860	. . . . . 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 3129	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
71		plyco0 26157	. . . . 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 585	. . . 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 257	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 ∘f + 𝐵) “ (ℤ≥‘(if(𝑀 ≤ 𝑁, 𝑁, 𝑀) + 1))) = {0})
74		simpl 482	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
75		simpr 484	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
76	13, 6	coeid 26203	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
77	76	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
78	16, 2	coeid 26203	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
79	78	adantl 481	. . . 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 26178	. . 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 26192	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘f + 𝐺)) = (𝐴 ∘f + 𝐵))
82		elfznn0 13574	. . . 4 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
83		ffvelcdm 7033	. . . 4 ⊢ (((𝐴 ∘f + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f + 𝐵)‘𝑘) ∈ ℂ)
84	22, 82, 83	syl2an 597	. . 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 26208	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
86	81, 85	jca 511	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 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429  ifcif 4466  {csn 4567   class class class wbr 5085   ↦ cmpt 5166   “ cima 5634  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   ≤ cle 11180  ℕ₀cn0 12437  ℤ_≥cuz 12788  ...cfz 13461  ↑cexp 14023  Σcsu 15648  Polycply 26149  coeffccoe 26151  degcdgr 26152

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-fl 13751  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-rlim 15451  df-sum 15649  df-0p 25637  df-ply 26153  df-coe 26155  df-dgr 26156

This theorem is referenced by:  coeadd  26216  dgradd  26232