Theorem dgrsub2 44095

Description: Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)

Hypothesis

Ref	Expression
dgrsub2.a	⊢ 𝑁 = (deg‘𝐹)

Assertion

Ref	Expression
dgrsub2	⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁)

Proof of Theorem dgrsub2

Step	Hyp	Ref	Expression
1		simpr2 1214	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ)
2		dgr0 26561	. . . . 5 ⊢ (deg‘0𝑝) = 0
3		nngt0 12350	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
4	2, 3	eqbrtrid 5140	. . . 4 ⊢ (𝑁 ∈ ℕ → (deg‘0𝑝) < 𝑁)
5		fveq2 6877	. . . . 5 ⊢ ((𝐹 ∘f − 𝐺) = 0𝑝 → (deg‘(𝐹 ∘f − 𝐺)) = (deg‘0𝑝))
6	5	breq1d 5113	. . . 4 ⊢ ((𝐹 ∘f − 𝐺) = 0𝑝 → ((deg‘(𝐹 ∘f − 𝐺)) < 𝑁 ↔ (deg‘0𝑝) < 𝑁))
7	4, 6	syl5ibrcom 250	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐹 ∘f − 𝐺) = 0𝑝 → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁))
8	1, 7	syl 18	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((𝐹 ∘f − 𝐺) = 0𝑝 → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁))
9		plyssc 26498	. . . . . . . 8 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
10	9	sseli 3927	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
11		plyssc 26498	. . . . . . . 8 ⊢ (Poly‘𝑇) ⊆ (Poly‘ℂ)
12	11	sseli 3927	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑇) → 𝐺 ∈ (Poly‘ℂ))
13		eqid 2761	. . . . . . . 8 ⊢ (deg‘𝐹) = (deg‘𝐹)
14		eqid 2761	. . . . . . . 8 ⊢ (deg‘𝐺) = (deg‘𝐺)
15	13, 14	dgrsub 26571	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (deg‘(𝐹 ∘f − 𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
16	10, 12, 15	syl2an 608	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (deg‘(𝐹 ∘f − 𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
17	16	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹 ∘f − 𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
18		simpr1 1213	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘𝐺) = 𝑁)
19		dgrsub2.a	. . . . . . . . 9 ⊢ 𝑁 = (deg‘𝐹)
20	19	eqcomi 2770	. . . . . . . 8 ⊢ (deg‘𝐹) = 𝑁
21	20	a1i 11	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘𝐹) = 𝑁)
22	18, 21	ifeq12d 4504	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)) = if((deg‘𝐹) ≤ (deg‘𝐺), 𝑁, 𝑁))
23		ifid 4523	. . . . . 6 ⊢ if((deg‘𝐹) ≤ (deg‘𝐺), 𝑁, 𝑁) = 𝑁
24	22, 23	eqtrdi 2812	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)) = 𝑁)
25	17, 24	breqtrd 5131	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹 ∘f − 𝐺)) ≤ 𝑁)
26		eqid 2761	. . . . . . . . 9 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
27		eqid 2761	. . . . . . . . 9 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
28	26, 27	coesub 26556	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘(𝐹 ∘f − 𝐺)) = ((coeff‘𝐹) ∘f − (coeff‘𝐺)))
29	10, 12, 28	syl2an 608	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (coeff‘(𝐹 ∘f − 𝐺)) = ((coeff‘𝐹) ∘f − (coeff‘𝐺)))
30	29	adantr 486	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘(𝐹 ∘f − 𝐺)) = ((coeff‘𝐹) ∘f − (coeff‘𝐺)))
31	30	fveq1d 6879	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘(𝐹 ∘f − 𝐺))‘𝑁) = (((coeff‘𝐹) ∘f − (coeff‘𝐺))‘𝑁))
32	1	nnnn0d 12648	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ0)
33	26	coef3 26531	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
34	33	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐹):ℕ0⟶ℂ)
35	34	ffnd 6702	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐹) Fn ℕ0)
36	27	coef3 26531	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑇) → (coeff‘𝐺):ℕ0⟶ℂ)
37	36	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐺):ℕ0⟶ℂ)
38	37	ffnd 6702	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐺) Fn ℕ0)
39		nn0ex 12593	. . . . . . . 8 ⊢ ℕ0 ∈ V
40	39	a1i 11	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ℕ0 ∈ V)
41		inidm 4172	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
42		simplr3 1236	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))
43		eqidd 2762	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐺)‘𝑁) = ((coeff‘𝐺)‘𝑁))
44	35, 38, 40, 40, 41, 42, 43	ofval 7693	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹) ∘f − (coeff‘𝐺))‘𝑁) = (((coeff‘𝐺)‘𝑁) − ((coeff‘𝐺)‘𝑁)))
45	32, 44	mpdan 700	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (((coeff‘𝐹) ∘f − (coeff‘𝐺))‘𝑁) = (((coeff‘𝐺)‘𝑁) − ((coeff‘𝐺)‘𝑁)))
46	37, 32	ffvelcdmd 7077	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
47	46	subidd 11638	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (((coeff‘𝐺)‘𝑁) − ((coeff‘𝐺)‘𝑁)) = 0)
48	31, 45, 47	3eqtrd 2800	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘(𝐹 ∘f − 𝐺))‘𝑁) = 0)
49		plysubcl 26521	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ))
50	10, 12, 49	syl2an 608	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ))
51	50	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ))
52		eqid 2761	. . . . . 6 ⊢ (deg‘(𝐹 ∘f − 𝐺)) = (deg‘(𝐹 ∘f − 𝐺))
53		eqid 2761	. . . . . 6 ⊢ (coeff‘(𝐹 ∘f − 𝐺)) = (coeff‘(𝐹 ∘f − 𝐺))
54	52, 53	dgrlt 26565	. . . . 5 ⊢ (((𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ) ∧ 𝑁 ∈ ℕ0) → (((𝐹 ∘f − 𝐺) = 0𝑝 ∨ (deg‘(𝐹 ∘f − 𝐺)) < 𝑁) ↔ ((deg‘(𝐹 ∘f − 𝐺)) ≤ 𝑁 ∧ ((coeff‘(𝐹 ∘f − 𝐺))‘𝑁) = 0)))
55	51, 32, 54	syl2anc 596	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (((𝐹 ∘f − 𝐺) = 0𝑝 ∨ (deg‘(𝐹 ∘f − 𝐺)) < 𝑁) ↔ ((deg‘(𝐹 ∘f − 𝐺)) ≤ 𝑁 ∧ ((coeff‘(𝐹 ∘f − 𝐺))‘𝑁) = 0)))
56	25, 48, 55	mpbir2and 726	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((𝐹 ∘f − 𝐺) = 0𝑝 ∨ (deg‘(𝐹 ∘f − 𝐺)) < 𝑁))
57	56	ord 878	. 2 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (¬ (𝐹 ∘f − 𝐺) = 0𝑝 → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁))
58	8, 57	pm2.61d 181	1 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ifcif 4482   class class class wbr 5103  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ∘_f cof 7680  ℂcc 11179  0cc0 11181   < clt 11324   ≤ cle 11325   − cmin 11522  ℕcn 12316  ℕ₀cn0 12587  0_𝑝c0p 25970  Polycply 26482  coeffccoe 26484  degcdgr 26485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-rlim 15636  df-sum 15834  df-0p 25971  df-ply 26486  df-coe 26488  df-dgr 26489

This theorem is used by:  mpaaeu  44110