Theorem dgradd2 26202

Description: The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
dgradd.1	⊢ 𝑀 = (deg‘𝐹)
dgradd.2	⊢ 𝑁 = (deg‘𝐺)

Assertion

Ref	Expression
dgradd2	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) = 𝑁)

Proof of Theorem dgradd2

Step	Hyp	Ref	Expression
1		plyaddcl 26153	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
2	1	3adant3 1132	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
3		dgrcl 26166	. . . . 5 ⊢ ((𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ) → (deg‘(𝐹 ∘f + 𝐺)) ∈ ℕ0)
4	2, 3	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ∈ ℕ0)
5	4	nn0red 12443	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ∈ ℝ)
6		dgradd.2	. . . . . . 7 ⊢ 𝑁 = (deg‘𝐺)
7		dgrcl 26166	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
8	6, 7	eqeltrid 2835	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
9	8	3ad2ant2 1134	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℕ0)
10	9	nn0red 12443	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℝ)
11		dgradd.1	. . . . . . 7 ⊢ 𝑀 = (deg‘𝐹)
12		dgrcl 26166	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
13	11, 12	eqeltrid 2835	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
14	13	3ad2ant1 1133	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℕ0)
15	14	nn0red 12443	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℝ)
16	10, 15	ifcld 4522	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ)
17	11, 6	dgradd 26201	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
18	17	3adant3 1132	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
19	10	leidd 11683	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≤ 𝑁)
20		simp3 1138	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 < 𝑁)
21	15, 10, 20	ltled 11261	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ≤ 𝑁)
22		breq1 5094	. . . . 5 ⊢ (𝑁 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀) → (𝑁 ≤ 𝑁 ↔ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁))
23		breq1 5094	. . . . 5 ⊢ (𝑀 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀) → (𝑀 ≤ 𝑁 ↔ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁))
24	22, 23	ifboth 4515	. . . 4 ⊢ ((𝑁 ≤ 𝑁 ∧ 𝑀 ≤ 𝑁) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁)
25	19, 21, 24	syl2anc 584	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁)
26	5, 16, 10, 18, 25	letrd 11270	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ≤ 𝑁)
27		eqid 2731	. . . . . . . 8 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
28		eqid 2731	. . . . . . . 8 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
29	27, 28	coeadd 26184	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘f + 𝐺)) = ((coeff‘𝐹) ∘f + (coeff‘𝐺)))
30	29	3adant3 1132	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘(𝐹 ∘f + 𝐺)) = ((coeff‘𝐹) ∘f + (coeff‘𝐺)))
31	30	fveq1d 6824	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) = (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁))
32	27	coef3 26165	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
33	32	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹):ℕ0⟶ℂ)
34	33	ffnd 6652	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹) Fn ℕ0)
35	28	coef3 26165	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
36	35	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺):ℕ0⟶ℂ)
37	36	ffnd 6652	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺) Fn ℕ0)
38		nn0ex 12387	. . . . . . . 8 ⊢ ℕ0 ∈ V
39	38	a1i 11	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ℕ0 ∈ V)
40		inidm 4177	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
41	15, 10	ltnled 11260	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝑀 < 𝑁 ↔ ¬ 𝑁 ≤ 𝑀))
42	20, 41	mpbid 232	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ¬ 𝑁 ≤ 𝑀)
43		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐹 ∈ (Poly‘𝑆))
44	27, 11	dgrub 26167	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝐹)‘𝑁) ≠ 0) → 𝑁 ≤ 𝑀)
45	44	3expia 1121	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹)‘𝑁) ≠ 0 → 𝑁 ≤ 𝑀))
46	43, 9, 45	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹)‘𝑁) ≠ 0 → 𝑁 ≤ 𝑀))
47	46	necon1bd 2946	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (¬ 𝑁 ≤ 𝑀 → ((coeff‘𝐹)‘𝑁) = 0))
48	42, 47	mpd 15	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐹)‘𝑁) = 0)
49	48	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹)‘𝑁) = 0)
50		eqidd 2732	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐺)‘𝑁) = ((coeff‘𝐺)‘𝑁))
51	34, 37, 39, 39, 40, 49, 50	ofval 7621	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
52	9, 51	mpdan 687	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
53	36, 9	ffvelcdmd 7018	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
54	53	addlidd 11314	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (0 + ((coeff‘𝐺)‘𝑁)) = ((coeff‘𝐺)‘𝑁))
55	31, 52, 54	3eqtrd 2770	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) = ((coeff‘𝐺)‘𝑁))
56		simp2 1137	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐺 ∈ (Poly‘𝑆))
57		0red 11115	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ∈ ℝ)
58	14	nn0ge0d 12445	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ≤ 𝑀)
59	57, 15, 10, 58, 20	lelttrd 11271	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 < 𝑁)
60	59	gt0ne0d 11681	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≠ 0)
61	6, 28	dgreq0 26199	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘𝑁) = 0))
62		fveq2 6822	. . . . . . . 8 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
63		dgr0 26196	. . . . . . . . 9 ⊢ (deg‘0𝑝) = 0
64	63	eqcomi 2740	. . . . . . . 8 ⊢ 0 = (deg‘0𝑝)
65	62, 6, 64	3eqtr4g 2791	. . . . . . 7 ⊢ (𝐺 = 0𝑝 → 𝑁 = 0)
66	61, 65	biimtrrdi 254	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (((coeff‘𝐺)‘𝑁) = 0 → 𝑁 = 0))
67	66	necon3d 2949	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝑁 ≠ 0 → ((coeff‘𝐺)‘𝑁) ≠ 0))
68	56, 60, 67	sylc 65	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ≠ 0)
69	55, 68	eqnetrd 2995	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) ≠ 0)
70		eqid 2731	. . . 4 ⊢ (coeff‘(𝐹 ∘f + 𝐺)) = (coeff‘(𝐹 ∘f + 𝐺))
71		eqid 2731	. . . 4 ⊢ (deg‘(𝐹 ∘f + 𝐺)) = (deg‘(𝐹 ∘f + 𝐺))
72	70, 71	dgrub 26167	. . 3 ⊢ (((𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) ≠ 0) → 𝑁 ≤ (deg‘(𝐹 ∘f + 𝐺)))
73	2, 9, 69, 72	syl3anc 1373	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≤ (deg‘(𝐹 ∘f + 𝐺)))
74	5, 10	letri3d 11255	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((deg‘(𝐹 ∘f + 𝐺)) = 𝑁 ↔ ((deg‘(𝐹 ∘f + 𝐺)) ≤ 𝑁 ∧ 𝑁 ≤ (deg‘(𝐹 ∘f + 𝐺)))))
75	26, 73, 74	mpbir2and 713	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) = 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ifcif 4475   class class class wbr 5091  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608  ℂcc 11004  ℝcr 11005  0cc0 11006   + caddc 11009   < clt 11146   ≤ cle 11147  ℕ₀cn0 12381  0_𝑝c0p 25598  Polycply 26117  coeffccoe 26119  degcdgr 26120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-rp 12891  df-fz 13408  df-fzo 13555  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-0p 25599  df-ply 26121  df-coe 26123  df-dgr 26124

This theorem is referenced by:  dgrcolem2  26208  plyremlem  26240  cjnpoly  46926