Theorem dgrsub2 42531

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 1193	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ)
2		dgr0 26190	. . . . 5 ⊢ (deg‘0𝑝) = 0
3		nngt0 12267	. . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
4	2, 3	eqbrtrid 5177	. . . 4 ⊢ (𝑁 ∈ ℕ → (deg‘0𝑝) < 𝑁)
5		fveq2 6891	. . . . 5 ⊢ ((𝐹 ∘f − 𝐺) = 0𝑝 → (deg‘(𝐹 ∘f − 𝐺)) = (deg‘0𝑝))
6	5	breq1d 5152	. . . 4 ⊢ ((𝐹 ∘f − 𝐺) = 0𝑝 → ((deg‘(𝐹 ∘f − 𝐺)) < 𝑁 ↔ (deg‘0𝑝) < 𝑁))
7	4, 6	syl5ibrcom 246	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐹 ∘f − 𝐺) = 0𝑝 → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁))
8	1, 7	syl 17	. 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 26127	. . . . . . . 8 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
10	9	sseli 3974	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
11		plyssc 26127	. . . . . . . 8 ⊢ (Poly‘𝑇) ⊆ (Poly‘ℂ)
12	11	sseli 3974	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑇) → 𝐺 ∈ (Poly‘ℂ))
13		eqid 2727	. . . . . . . 8 ⊢ (deg‘𝐹) = (deg‘𝐹)
14		eqid 2727	. . . . . . . 8 ⊢ (deg‘𝐺) = (deg‘𝐺)
15	13, 14	dgrsub 26200	. . . . . . 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 595	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (deg‘(𝐹 ∘f − 𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
17	16	adantr 480	. . . . 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 1192	. . . . . . 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 2736	. . . . . . . 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 4545	. . . . . 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 4564	. . . . . 6 ⊢ if((deg‘𝐹) ≤ (deg‘𝐺), 𝑁, 𝑁) = 𝑁
24	22, 23	eqtrdi 2783	. . . . 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 5168	. . . 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 2727	. . . . . . . . 9 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
27		eqid 2727	. . . . . . . . 9 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
28	26, 27	coesub 26184	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘(𝐹 ∘f − 𝐺)) = ((coeff‘𝐹) ∘f − (coeff‘𝐺)))
29	10, 12, 28	syl2an 595	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (coeff‘(𝐹 ∘f − 𝐺)) = ((coeff‘𝐹) ∘f − (coeff‘𝐺)))
30	29	adantr 480	. . . . . 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 6893	. . . . 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 12556	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ0)
33	26	coef3 26159	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
34	33	ad2antrr 725	. . . . . . . 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 6717	. . . . . . 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 26159	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑇) → (coeff‘𝐺):ℕ0⟶ℂ)
37	36	ad2antlr 726	. . . . . . . 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 6717	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐺) Fn ℕ0)
39		nn0ex 12502	. . . . . . . 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 4214	. . . . . . 7 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
42		simplr3 1215	. . . . . . 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 2728	. . . . . . 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 7690	. . . . . 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 686	. . . . 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 7089	. . . . . 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 11583	. . . . 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 2771	. . . 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 26149	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ))
50	10, 12, 49	syl2an 595	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ))
51	50	adantr 480	. . . . 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 2727	. . . . . 6 ⊢ (deg‘(𝐹 ∘f − 𝐺)) = (deg‘(𝐹 ∘f − 𝐺))
53		eqid 2727	. . . . . 6 ⊢ (coeff‘(𝐹 ∘f − 𝐺)) = (coeff‘(𝐹 ∘f − 𝐺))
54	52, 53	dgrlt 26194	. . . . 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 583	. . . 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 712	. . 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 863	. 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 179	1 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3469  ifcif 4524   class class class wbr 5142  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ∘_f cof 7677  ℂcc 11130  0cc0 11132   < clt 11272   ≤ cle 11273   − cmin 11468  ℕcn 12236  ℕ₀cn0 12496  0_𝑝c0p 25591  Polycply 26111  coeffccoe 26113  degcdgr 26114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9459  df-inf 9460  df-oi 9527  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-rp 13001  df-fz 13511  df-fzo 13654  df-fl 13783  df-seq 13993  df-exp 14053  df-hash 14316  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15458  df-rlim 15459  df-sum 15659  df-0p 25592  df-ply 26115  df-coe 26117  df-dgr 26118

This theorem is referenced by:  mpaaeu  42546