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Theorem dgradd2 26121
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 = (degβ€˜πΉ)
dgradd.2 𝑁 = (degβ€˜πΊ)
Assertion
Ref Expression
dgradd2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) = 𝑁)
Proof of Theorem dgradd2
StepHypRef Expression
1 plyaddcl 26072 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f + 𝐺) ∈ (Polyβ€˜β„‚))
213adant3 1131 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (𝐹 ∘f + 𝐺) ∈ (Polyβ€˜β„‚))
3 dgrcl 26085 . . . . 5 ((𝐹 ∘f + 𝐺) ∈ (Polyβ€˜β„‚) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ∈ β„•0)
42, 3syl 17 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ∈ β„•0)
54nn0red 12540 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ∈ ℝ)
6 dgradd.2 . . . . . . 7 𝑁 = (degβ€˜πΊ)
7 dgrcl 26085 . . . . . . 7 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΊ) ∈ β„•0)
86, 7eqeltrid 2836 . . . . . 6 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝑁 ∈ β„•0)
983ad2ant2 1133 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ∈ β„•0)
109nn0red 12540 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ∈ ℝ)
11 dgradd.1 . . . . . . 7 𝑀 = (degβ€˜πΉ)
12 dgrcl 26085 . . . . . . 7 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΉ) ∈ β„•0)
1311, 12eqeltrid 2836 . . . . . 6 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝑀 ∈ β„•0)
14133ad2ant1 1132 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 ∈ β„•0)
1514nn0red 12540 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 ∈ ℝ)
1610, 15ifcld 4574 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ ℝ)
1711, 6dgradd 26120 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
18173adant3 1131 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
1910leidd 11787 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ≀ 𝑁)
20 simp3 1137 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 < 𝑁)
2115, 10, 20ltled 11369 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 ≀ 𝑁)
22 breq1 5151 . . . . 5 (𝑁 = if(𝑀 ≀ 𝑁, 𝑁, 𝑀) β†’ (𝑁 ≀ 𝑁 ↔ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁))
23 breq1 5151 . . . . 5 (𝑀 = if(𝑀 ≀ 𝑁, 𝑁, 𝑀) β†’ (𝑀 ≀ 𝑁 ↔ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁))
2422, 23ifboth 4567 . . . 4 ((𝑁 ≀ 𝑁 ∧ 𝑀 ≀ 𝑁) β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁)
2519, 21, 24syl2anc 583 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁)
265, 16, 10, 18, 25letrd 11378 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ≀ 𝑁)
27 eqid 2731 . . . . . . . 8 (coeffβ€˜πΉ) = (coeffβ€˜πΉ)
28 eqid 2731 . . . . . . . 8 (coeffβ€˜πΊ) = (coeffβ€˜πΊ)
2927, 28coeadd 26103 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (coeffβ€˜(𝐹 ∘f + 𝐺)) = ((coeffβ€˜πΉ) ∘f + (coeffβ€˜πΊ)))
30293adant3 1131 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜(𝐹 ∘f + 𝐺)) = ((coeffβ€˜πΉ) ∘f + (coeffβ€˜πΊ)))
3130fveq1d 6893 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜(𝐹 ∘f + 𝐺))β€˜π‘) = (((coeffβ€˜πΉ) ∘f + (coeffβ€˜πΊ))β€˜π‘))
3227coef3 26084 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„‚)
33323ad2ant1 1132 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„‚)
3433ffnd 6718 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΉ) Fn β„•0)
3528coef3 26084 . . . . . . . . 9 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (coeffβ€˜πΊ):β„•0βŸΆβ„‚)
36353ad2ant2 1133 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΊ):β„•0βŸΆβ„‚)
3736ffnd 6718 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΊ) Fn β„•0)
38 nn0ex 12485 . . . . . . . 8 β„•0 ∈ V
3938a1i 11 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ β„•0 ∈ V)
40 inidm 4218 . . . . . . 7 (β„•0 ∩ β„•0) = β„•0
4115, 10ltnled 11368 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (𝑀 < 𝑁 ↔ Β¬ 𝑁 ≀ 𝑀))
4220, 41mpbid 231 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ Β¬ 𝑁 ≀ 𝑀)
43 simp1 1135 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝐹 ∈ (Polyβ€˜π‘†))
4427, 11dgrub 26086 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝑁 ∈ β„•0 ∧ ((coeffβ€˜πΉ)β€˜π‘) β‰  0) β†’ 𝑁 ≀ 𝑀)
45443expia 1120 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝑁 ∈ β„•0) β†’ (((coeffβ€˜πΉ)β€˜π‘) β‰  0 β†’ 𝑁 ≀ 𝑀))
4643, 9, 45syl2anc 583 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (((coeffβ€˜πΉ)β€˜π‘) β‰  0 β†’ 𝑁 ≀ 𝑀))
4746necon1bd 2957 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (Β¬ 𝑁 ≀ 𝑀 β†’ ((coeffβ€˜πΉ)β€˜π‘) = 0))
4842, 47mpd 15 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜πΉ)β€˜π‘) = 0)
4948adantr 480 . . . . . . 7 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ β„•0) β†’ ((coeffβ€˜πΉ)β€˜π‘) = 0)
50 eqidd 2732 . . . . . . 7 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ β„•0) β†’ ((coeffβ€˜πΊ)β€˜π‘) = ((coeffβ€˜πΊ)β€˜π‘))
5134, 37, 39, 39, 40, 49, 50ofval 7685 . . . . . 6 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ β„•0) β†’ (((coeffβ€˜πΉ) ∘f + (coeffβ€˜πΊ))β€˜π‘) = (0 + ((coeffβ€˜πΊ)β€˜π‘)))
529, 51mpdan 684 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (((coeffβ€˜πΉ) ∘f + (coeffβ€˜πΊ))β€˜π‘) = (0 + ((coeffβ€˜πΊ)β€˜π‘)))
5336, 9ffvelcdmd 7087 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜πΊ)β€˜π‘) ∈ β„‚)
5453addlidd 11422 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (0 + ((coeffβ€˜πΊ)β€˜π‘)) = ((coeffβ€˜πΊ)β€˜π‘))
5531, 52, 543eqtrd 2775 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜(𝐹 ∘f + 𝐺))β€˜π‘) = ((coeffβ€˜πΊ)β€˜π‘))
56 simp2 1136 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝐺 ∈ (Polyβ€˜π‘†))
57 0red 11224 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 0 ∈ ℝ)
5814nn0ge0d 12542 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 0 ≀ 𝑀)
5957, 15, 10, 58, 20lelttrd 11379 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 0 < 𝑁)
6059gt0ne0d 11785 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 β‰  0)
616, 28dgreq0 26118 . . . . . . 7 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (𝐺 = 0𝑝 ↔ ((coeffβ€˜πΊ)β€˜π‘) = 0))
62 fveq2 6891 . . . . . . . 8 (𝐺 = 0𝑝 β†’ (degβ€˜πΊ) = (degβ€˜0𝑝))
63 dgr0 26115 . . . . . . . . 9 (degβ€˜0𝑝) = 0
6463eqcomi 2740 . . . . . . . 8 0 = (degβ€˜0𝑝)
6562, 6, 643eqtr4g 2796 . . . . . . 7 (𝐺 = 0𝑝 β†’ 𝑁 = 0)
6661, 65syl6bir 254 . . . . . 6 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (((coeffβ€˜πΊ)β€˜π‘) = 0 β†’ 𝑁 = 0))
6766necon3d 2960 . . . . 5 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (𝑁 β‰  0 β†’ ((coeffβ€˜πΊ)β€˜π‘) β‰  0))
6856, 60, 67sylc 65 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜πΊ)β€˜π‘) β‰  0)
6955, 68eqnetrd 3007 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜(𝐹 ∘f + 𝐺))β€˜π‘) β‰  0)
70 eqid 2731 . . . 4 (coeffβ€˜(𝐹 ∘f + 𝐺)) = (coeffβ€˜(𝐹 ∘f + 𝐺))
71 eqid 2731 . . . 4 (degβ€˜(𝐹 ∘f + 𝐺)) = (degβ€˜(𝐹 ∘f + 𝐺))
7270, 71dgrub 26086 . . 3 (((𝐹 ∘f + 𝐺) ∈ (Polyβ€˜β„‚) ∧ 𝑁 ∈ β„•0 ∧ ((coeffβ€˜(𝐹 ∘f + 𝐺))β€˜π‘) β‰  0) β†’ 𝑁 ≀ (degβ€˜(𝐹 ∘f + 𝐺)))
732, 9, 69, 72syl3anc 1370 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ≀ (degβ€˜(𝐹 ∘f + 𝐺)))
745, 10letri3d 11363 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((degβ€˜(𝐹 ∘f + 𝐺)) = 𝑁 ↔ ((degβ€˜(𝐹 ∘f + 𝐺)) ≀ 𝑁 ∧ 𝑁 ≀ (degβ€˜(𝐹 ∘f + 𝐺)))))
7526, 73, 74mpbir2and 710 1 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  Vcvv 3473  ifcif 4528   class class class wbr 5148  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ∘f cof 7672  β„‚cc 11114  β„cr 11115  0cc0 11116   + caddc 11119   < clt 11255   ≀ cle 11256  β„•0cn0 12479  0𝑝c0p 25518  Polycply 26036  coeffccoe 26038  degcdgr 26039
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-fz 13492  df-fzo 13635  df-fl 13764  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-rlim 15440  df-sum 15640  df-0p 25519  df-ply 26040  df-coe 26042  df-dgr 26043
This theorem is referenced by:  dgrcolem2  26127  plyremlem  26156
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