Theorem coeaddlem 26282

Description: Lemma for coeadd 26284 and dgradd 26300. (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 26253	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
2		coeadd.4	. . . . . 6 ⊢ 𝑁 = (deg‘𝐺)
3		dgrcl 26266	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
4	2, 3	eqeltrid 2860	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
5	4	adantl 484	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6		coeadd.3	. . . . . 6 ⊢ 𝑀 = (deg‘𝐹)
7		dgrcl 26266	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
8	6, 7	eqeltrid 2860	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
9	8	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
10	5, 9	ifcld 4521	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
11		addcl 11145	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
12	11	adantl 484	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13		coefv0.1	. . . . . 6 ⊢ 𝐴 = (coeff‘𝐹)
14	13	coef3 26265	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
15	14	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16		coeadd.2	. . . . . 6 ⊢ 𝐵 = (coeff‘𝐺)
17	16	coef3 26265	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
18	17	adantl 484	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19		nn0ex 12477	. . . . 5 ⊢ ℕ0 ∈ V
20	19	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21		inidm 4173	. . . 4 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
22	12, 15, 18, 20, 20, 21	off 7667	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘f + 𝐵):ℕ0⟶ℂ)
23		oveq12 7394	. . . . . . . . . 10 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = (0 + 0))
24		00id 11348	. . . . . . . . . 10 ⊢ (0 + 0) = 0
25	23, 24	eqtrdi 2807	. . . . . . . . 9 ⊢ (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0)
26	15	ffnd 6681	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
27	18	ffnd 6681	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
28		eqidd 2757	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐴‘𝑘))
29		eqidd 2757	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵‘𝑘) = (𝐵‘𝑘))
30	26, 27, 20, 20, 21, 28, 29	ofval 7660	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f + 𝐵)‘𝑘) = ((𝐴‘𝑘) + (𝐵‘𝑘)))
31	30	eqeq1d 2758	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) = 0 ↔ ((𝐴‘𝑘) + (𝐵‘𝑘)) = 0))
32	25, 31	imbitrrid 248	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0) → ((𝐴 ∘f + 𝐵)‘𝑘) = 0))
33	32	necon3ad 2964	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0)))
34		neorian 3046	. . . . . . 7 ⊢ (((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0) ↔ ¬ ((𝐴‘𝑘) = 0 ∧ (𝐵‘𝑘) = 0))
35	33, 34	imbitrrdi 254	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → ((𝐴‘𝑘) ≠ 0 ∨ (𝐵‘𝑘) ≠ 0)))
36	13, 6	dgrub2 26268	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
37	36	adantr 483	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
38		plyco0 26225	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
39	9, 15, 38	syl2anc 592	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)))
40	37, 39	mpbid 234	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
41	40	r19.21bi 3248	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
42	9	adantr 483	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
43	42	nn0red 12533	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
44	5	adantr 483	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
45	44	nn0red 12533	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
46		max1 13178	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
47	43, 45, 46	syl2anc 592	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
48		nn0re 12480	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
49	48	adantl 484	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
50	10	adantr 483	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
51	50	nn0red 12533	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ)
52		letr 11267	. . . . . . . . . 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 1386	. . . . . . . . 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 702	. . . . . . . 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 26268	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
57	56	adantl 484	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})
58		plyco0 26225	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
59	5, 18, 58	syl2anc 592	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
60	57, 59	mpbid 234	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
61	60	r19.21bi 3248	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
62		max2 13180	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
63	43, 45, 62	syl2anc 592	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
64		letr 11267	. . . . . . . . . 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 1386	. . . . . . . . 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 702	. . . . . . . 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 868	. . . . . 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 3148	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴 ∘f + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
71		plyco0 26225	. . . . 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 592	. . . 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 259	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 ∘f + 𝐵) “ (ℤ≥‘(if(𝑀 ≤ 𝑁, 𝑁, 𝑀) + 1))) = {0})
74		simpl 485	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
75		simpr 487	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
76	13, 6	coeid 26271	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
77	76	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
78	16, 2	coeid 26271	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
79	78	adantl 484	. . . 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 26246	. . 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 26260	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘f + 𝐺)) = (𝐴 ∘f + 𝐵))
82		elfznn0 13615	. . . 4 ⊢ (𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
83		ffvelcdm 7051	. . . 4 ⊢ (((𝐴 ∘f + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴 ∘f + 𝐵)‘𝑘) ∈ ℂ)
84	22, 82, 83	syl2an 604	. . 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 26276	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
86	81, 85	jca 518	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 208   ∧ wa 398   ∨ wo 856   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  ∀wral 3070  Vcvv 3448  ifcif 4474  {csn 4576   class class class wbr 5094   ↦ cmpt 5175   “ cima 5643  ⟶wf 6506  ‘cfv 6510  (class class class)co 7385   ∘_f cof 7647  ℂcc 11061  ℝcr 11062  0cc0 11063  1c1 11064   + caddc 11066   · cmul 11068   ≤ cle 11207  ℕ₀cn0 12471  ℤ_≥cuz 12829  ...cfz 13502  ↑cexp 14064  Σcsu 15689  Polycply 26217  coeffccoe 26219  degcdgr 26220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-inf2 9586  ax-cnex 11119  ax-resscn 11120  ax-1cn 11121  ax-icn 11122  ax-addcl 11123  ax-addrcl 11124  ax-mulcl 11125  ax-mulrcl 11126  ax-mulcom 11127  ax-addass 11128  ax-mulass 11129  ax-distr 11130  ax-i2m1 11131  ax-1ne0 11132  ax-1rid 11133  ax-rnegex 11134  ax-rrecex 11135  ax-cnre 11136  ax-pre-lttri 11137  ax-pre-lttrn 11138  ax-pre-ltadd 11139  ax-pre-mulgt0 11140  ax-pre-sup 11141

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-nel 3056  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-isom 6519  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-of 7649  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-er 8666  df-map 8798  df-pm 8799  df-en 8917  df-dom 8918  df-sdom 8919  df-fin 8920  df-sup 9378  df-inf 9379  df-oi 9448  df-card 9887  df-pnf 11208  df-mnf 11209  df-xr 11210  df-ltxr 11211  df-le 11212  df-sub 11406  df-neg 11407  df-div 11835  df-nn 12201  df-2 12270  df-3 12271  df-n0 12472  df-z 12559  df-uz 12830  df-rp 12984  df-fz 13503  df-fzo 13650  df-fl 13792  df-seq 14005  df-exp 14065  df-hash 14334  df-cj 15102  df-re 15103  df-im 15104  df-sqrt 15238  df-abs 15239  df-clim 15491  df-rlim 15492  df-sum 15690  df-0p 25705  df-ply 26221  df-coe 26223  df-dgr 26224

This theorem is referenced by:  coeadd  26284  dgradd  26300