Theorem dgradd2 25666

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 25618	. . . . . 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 25631	. . . . 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 12483	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ∈ ℝ)
6		dgradd.2	. . . . . . 7 ⊢ 𝑁 = (deg‘𝐺)
7		dgrcl 25631	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
8	6, 7	eqeltrid 2836	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
9	8	3ad2ant2 1134	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℕ0)
10	9	nn0red 12483	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℝ)
11		dgradd.1	. . . . . . 7 ⊢ 𝑀 = (deg‘𝐹)
12		dgrcl 25631	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
13	11, 12	eqeltrid 2836	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
14	13	3ad2ant1 1133	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℕ0)
15	14	nn0red 12483	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℝ)
16	10, 15	ifcld 4537	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ)
17	11, 6	dgradd 25665	. . . 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 11730	. . . 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 11312	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ≤ 𝑁)
22		breq1 5113	. . . . 5 ⊢ (𝑁 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀) → (𝑁 ≤ 𝑁 ↔ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁))
23		breq1 5113	. . . . 5 ⊢ (𝑀 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀) → (𝑀 ≤ 𝑁 ↔ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁))
24	22, 23	ifboth 4530	. . . 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 11321	. 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 25649	. . . . . . 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 6849	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) = (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁))
32	27	coef3 25630	. . . . . . . . 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 6674	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹) Fn ℕ0)
35	28	coef3 25630	. . . . . . . . 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 6674	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺) Fn ℕ0)
38		nn0ex 12428	. . . . . . . 8 ⊢ ℕ0 ∈ V
39	38	a1i 11	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ℕ0 ∈ V)
40		inidm 4183	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
41	15, 10	ltnled 11311	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝑀 < 𝑁 ↔ ¬ 𝑁 ≤ 𝑀))
42	20, 41	mpbid 231	. . . . . . . . 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 25632	. . . . . . . . . . . 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 2957	. . . . . . . . 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 481	. . . . . . 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 7633	. . . . . 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 685	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
53	36, 9	ffvelcdmd 7041	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
54	53	addlidd 11365	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (0 + ((coeff‘𝐺)‘𝑁)) = ((coeff‘𝐺)‘𝑁))
55	31, 52, 54	3eqtrd 2775	. . . 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 11167	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ∈ ℝ)
58	14	nn0ge0d 12485	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ≤ 𝑀)
59	57, 15, 10, 58, 20	lelttrd 11322	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 < 𝑁)
60	59	gt0ne0d 11728	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≠ 0)
61	6, 28	dgreq0 25663	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘𝑁) = 0))
62		fveq2 6847	. . . . . . . 8 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
63		dgr0 25660	. . . . . . . . 9 ⊢ (deg‘0𝑝) = 0
64	63	eqcomi 2740	. . . . . . . 8 ⊢ 0 = (deg‘0𝑝)
65	62, 6, 64	3eqtr4g 2796	. . . . . . 7 ⊢ (𝐺 = 0𝑝 → 𝑁 = 0)
66	61, 65	syl6bir 253	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (((coeff‘𝐺)‘𝑁) = 0 → 𝑁 = 0))
67	66	necon3d 2960	. . . . 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 3007	. . 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 25632	. . 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 1371	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≤ (deg‘(𝐹 ∘f + 𝐺)))
74	5, 10	letri3d 11306	. 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 711	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) = 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  Vcvv 3446  ifcif 4491   class class class wbr 5110  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362   ∘_f cof 7620  ℂcc 11058  ℝcr 11059  0cc0 11060   + caddc 11063   < clt 11198   ≤ cle 11199  ℕ₀cn0 12422  0_𝑝c0p 25070  Polycply 25582  coeffccoe 25584  degcdgr 25585

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9387  df-inf 9388  df-oi 9455  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-fz 13435  df-fzo 13578  df-fl 13707  df-seq 13917  df-exp 13978  df-hash 14241  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-clim 15382  df-rlim 15383  df-sum 15583  df-0p 25071  df-ply 25586  df-coe 25588  df-dgr 25589

This theorem is referenced by:  dgrcolem2  25672  plyremlem  25701