Theorem dgradd2 25439

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 25391	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
2	1	3adant3 1131	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
3		dgrcl 25404	. . . . 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 12304	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ∈ ℝ)
6		dgradd.2	. . . . . . 7 ⊢ 𝑁 = (deg‘𝐺)
7		dgrcl 25404	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
8	6, 7	eqeltrid 2843	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
9	8	3ad2ant2 1133	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℕ0)
10	9	nn0red 12304	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℝ)
11		dgradd.1	. . . . . . 7 ⊢ 𝑀 = (deg‘𝐹)
12		dgrcl 25404	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
13	11, 12	eqeltrid 2843	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
14	13	3ad2ant1 1132	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℕ0)
15	14	nn0red 12304	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℝ)
16	10, 15	ifcld 4505	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℝ)
17	11, 6	dgradd 25438	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
18	17	3adant3 1131	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
19	10	leidd 11551	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≤ 𝑁)
20		simp3 1137	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 < 𝑁)
21	15, 10, 20	ltled 11133	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ≤ 𝑁)
22		breq1 5076	. . . . 5 ⊢ (𝑁 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀) → (𝑁 ≤ 𝑁 ↔ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁))
23		breq1 5076	. . . . 5 ⊢ (𝑀 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀) → (𝑀 ≤ 𝑁 ↔ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ≤ 𝑁))
24	22, 23	ifboth 4498	. . . 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 11142	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) ≤ 𝑁)
27		eqid 2738	. . . . . . . 8 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
28		eqid 2738	. . . . . . . 8 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
29	27, 28	coeadd 25422	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘f + 𝐺)) = ((coeff‘𝐹) ∘f + (coeff‘𝐺)))
30	29	3adant3 1131	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘(𝐹 ∘f + 𝐺)) = ((coeff‘𝐹) ∘f + (coeff‘𝐺)))
31	30	fveq1d 6768	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) = (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁))
32	27	coef3 25403	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
33	32	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹):ℕ0⟶ℂ)
34	33	ffnd 6593	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹) Fn ℕ0)
35	28	coef3 25403	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
36	35	3ad2ant2 1133	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺):ℕ0⟶ℂ)
37	36	ffnd 6593	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺) Fn ℕ0)
38		nn0ex 12249	. . . . . . . 8 ⊢ ℕ0 ∈ V
39	38	a1i 11	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ℕ0 ∈ V)
40		inidm 4152	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
41	15, 10	ltnled 11132	. . . . . . . . . 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 1135	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐹 ∈ (Poly‘𝑆))
44	27, 11	dgrub 25405	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝐹)‘𝑁) ≠ 0) → 𝑁 ≤ 𝑀)
45	44	3expia 1120	. . . . . . . . . . 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 2961	. . . . . . . . 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 2739	. . . . . . 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 7534	. . . . . 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 684	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
53	36, 9	ffvelrnd 6954	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
54	53	addid2d 11186	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (0 + ((coeff‘𝐺)‘𝑁)) = ((coeff‘𝐺)‘𝑁))
55	31, 52, 54	3eqtrd 2782	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) = ((coeff‘𝐺)‘𝑁))
56		simp2 1136	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐺 ∈ (Poly‘𝑆))
57		0red 10988	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ∈ ℝ)
58	14	nn0ge0d 12306	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ≤ 𝑀)
59	57, 15, 10, 58, 20	lelttrd 11143	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 < 𝑁)
60	59	gt0ne0d 11549	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≠ 0)
61	6, 28	dgreq0 25436	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘𝑁) = 0))
62		fveq2 6766	. . . . . . . 8 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
63		dgr0 25433	. . . . . . . . 9 ⊢ (deg‘0𝑝) = 0
64	63	eqcomi 2747	. . . . . . . 8 ⊢ 0 = (deg‘0𝑝)
65	62, 6, 64	3eqtr4g 2803	. . . . . . 7 ⊢ (𝐺 = 0𝑝 → 𝑁 = 0)
66	61, 65	syl6bir 253	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (((coeff‘𝐺)‘𝑁) = 0 → 𝑁 = 0))
67	66	necon3d 2964	. . . . 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 3011	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹 ∘f + 𝐺))‘𝑁) ≠ 0)
70		eqid 2738	. . . 4 ⊢ (coeff‘(𝐹 ∘f + 𝐺)) = (coeff‘(𝐹 ∘f + 𝐺))
71		eqid 2738	. . . 4 ⊢ (deg‘(𝐹 ∘f + 𝐺)) = (deg‘(𝐹 ∘f + 𝐺))
72	70, 71	dgrub 25405	. . 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 1370	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≤ (deg‘(𝐹 ∘f + 𝐺)))
74	5, 10	letri3d 11127	. 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 710	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹 ∘f + 𝐺)) = 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3429  ifcif 4459   class class class wbr 5073  ⟶wf 6422  ‘cfv 6426  (class class class)co 7267   ∘_f cof 7521  ℂcc 10879  ℝcr 10880  0cc0 10881   + caddc 10884   < clt 11019   ≤ cle 11020  ℕ₀cn0 12243  0_𝑝c0p 24843  Polycply 25355  coeffccoe 25357  degcdgr 25358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-inf2 9386  ax-cnex 10937  ax-resscn 10938  ax-1cn 10939  ax-icn 10940  ax-addcl 10941  ax-addrcl 10942  ax-mulcl 10943  ax-mulrcl 10944  ax-mulcom 10945  ax-addass 10946  ax-mulass 10947  ax-distr 10948  ax-i2m1 10949  ax-1ne0 10950  ax-1rid 10951  ax-rnegex 10952  ax-rrecex 10953  ax-cnre 10954  ax-pre-lttri 10955  ax-pre-lttrn 10956  ax-pre-ltadd 10957  ax-pre-mulgt0 10958  ax-pre-sup 10959

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-se 5540  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-isom 6435  df-riota 7224  df-ov 7270  df-oprab 7271  df-mpo 7272  df-of 7523  df-om 7703  df-1st 7820  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-1o 8284  df-er 8485  df-map 8604  df-pm 8605  df-en 8721  df-dom 8722  df-sdom 8723  df-fin 8724  df-sup 9188  df-inf 9189  df-oi 9256  df-card 9707  df-pnf 11021  df-mnf 11022  df-xr 11023  df-ltxr 11024  df-le 11025  df-sub 11217  df-neg 11218  df-div 11643  df-nn 11984  df-2 12046  df-3 12047  df-n0 12244  df-z 12330  df-uz 12593  df-rp 12741  df-fz 13250  df-fzo 13393  df-fl 13522  df-seq 13732  df-exp 13793  df-hash 14055  df-cj 14820  df-re 14821  df-im 14822  df-sqrt 14956  df-abs 14957  df-clim 15207  df-rlim 15208  df-sum 15408  df-0p 24844  df-ply 25359  df-coe 25361  df-dgr 25362

This theorem is referenced by:  dgrcolem2  25445  plyremlem  25474