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Theorem dgradd2 25782
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 25734 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f + 𝐺) ∈ (Polyβ€˜β„‚))
213adant3 1133 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (𝐹 ∘f + 𝐺) ∈ (Polyβ€˜β„‚))
3 dgrcl 25747 . . . . 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 12533 . . 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 25747 . . . . . . 7 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΊ) ∈ β„•0)
86, 7eqeltrid 2838 . . . . . 6 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝑁 ∈ β„•0)
983ad2ant2 1135 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ∈ β„•0)
109nn0red 12533 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ∈ ℝ)
11 dgradd.1 . . . . . . 7 𝑀 = (degβ€˜πΉ)
12 dgrcl 25747 . . . . . . 7 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΉ) ∈ β„•0)
1311, 12eqeltrid 2838 . . . . . 6 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝑀 ∈ β„•0)
14133ad2ant1 1134 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 ∈ β„•0)
1514nn0red 12533 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 ∈ ℝ)
1610, 15ifcld 4575 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ ℝ)
1711, 6dgradd 25781 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
18173adant3 1133 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
1910leidd 11780 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ≀ 𝑁)
20 simp3 1139 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 < 𝑁)
2115, 10, 20ltled 11362 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑀 ≀ 𝑁)
22 breq1 5152 . . . . 5 (𝑁 = if(𝑀 ≀ 𝑁, 𝑁, 𝑀) β†’ (𝑁 ≀ 𝑁 ↔ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁))
23 breq1 5152 . . . . 5 (𝑀 = if(𝑀 ≀ 𝑁, 𝑁, 𝑀) β†’ (𝑀 ≀ 𝑁 ↔ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁))
2422, 23ifboth 4568 . . . 4 ((𝑁 ≀ 𝑁 ∧ 𝑀 ≀ 𝑁) β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁)
2519, 21, 24syl2anc 585 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ≀ 𝑁)
265, 16, 10, 18, 25letrd 11371 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (degβ€˜(𝐹 ∘f + 𝐺)) ≀ 𝑁)
27 eqid 2733 . . . . . . . 8 (coeffβ€˜πΉ) = (coeffβ€˜πΉ)
28 eqid 2733 . . . . . . . 8 (coeffβ€˜πΊ) = (coeffβ€˜πΊ)
2927, 28coeadd 25765 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (coeffβ€˜(𝐹 ∘f + 𝐺)) = ((coeffβ€˜πΉ) ∘f + (coeffβ€˜πΊ)))
30293adant3 1133 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜(𝐹 ∘f + 𝐺)) = ((coeffβ€˜πΉ) ∘f + (coeffβ€˜πΊ)))
3130fveq1d 6894 . . . . 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 25746 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„‚)
33323ad2ant1 1134 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΉ):β„•0βŸΆβ„‚)
3433ffnd 6719 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΉ) Fn β„•0)
3528coef3 25746 . . . . . . . . 9 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (coeffβ€˜πΊ):β„•0βŸΆβ„‚)
36353ad2ant2 1135 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΊ):β„•0βŸΆβ„‚)
3736ffnd 6719 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (coeffβ€˜πΊ) Fn β„•0)
38 nn0ex 12478 . . . . . . . 8 β„•0 ∈ V
3938a1i 11 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ β„•0 ∈ V)
40 inidm 4219 . . . . . . 7 (β„•0 ∩ β„•0) = β„•0
4115, 10ltnled 11361 . . . . . . . . . 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 1137 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝐹 ∈ (Polyβ€˜π‘†))
4427, 11dgrub 25748 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝑁 ∈ β„•0 ∧ ((coeffβ€˜πΉ)β€˜π‘) β‰  0) β†’ 𝑁 ≀ 𝑀)
45443expia 1122 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝑁 ∈ β„•0) β†’ (((coeffβ€˜πΉ)β€˜π‘) β‰  0 β†’ 𝑁 ≀ 𝑀))
4643, 9, 45syl2anc 585 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (((coeffβ€˜πΉ)β€˜π‘) β‰  0 β†’ 𝑁 ≀ 𝑀))
4746necon1bd 2959 . . . . . . . . 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 482 . . . . . . 7 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ β„•0) β†’ ((coeffβ€˜πΉ)β€˜π‘) = 0)
50 eqidd 2734 . . . . . . 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 7681 . . . . . 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 686 . . . . 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 7088 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜πΊ)β€˜π‘) ∈ β„‚)
5453addlidd 11415 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ (0 + ((coeffβ€˜πΊ)β€˜π‘)) = ((coeffβ€˜πΊ)β€˜π‘))
5531, 52, 543eqtrd 2777 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜(𝐹 ∘f + 𝐺))β€˜π‘) = ((coeffβ€˜πΊ)β€˜π‘))
56 simp2 1138 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝐺 ∈ (Polyβ€˜π‘†))
57 0red 11217 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 0 ∈ ℝ)
5814nn0ge0d 12535 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 0 ≀ 𝑀)
5957, 15, 10, 58, 20lelttrd 11372 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 0 < 𝑁)
6059gt0ne0d 11778 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 β‰  0)
616, 28dgreq0 25779 . . . . . . 7 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (𝐺 = 0𝑝 ↔ ((coeffβ€˜πΊ)β€˜π‘) = 0))
62 fveq2 6892 . . . . . . . 8 (𝐺 = 0𝑝 β†’ (degβ€˜πΊ) = (degβ€˜0𝑝))
63 dgr0 25776 . . . . . . . . 9 (degβ€˜0𝑝) = 0
6463eqcomi 2742 . . . . . . . 8 0 = (degβ€˜0𝑝)
6562, 6, 643eqtr4g 2798 . . . . . . 7 (𝐺 = 0𝑝 β†’ 𝑁 = 0)
6661, 65syl6bir 254 . . . . . 6 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (((coeffβ€˜πΊ)β€˜π‘) = 0 β†’ 𝑁 = 0))
6766necon3d 2962 . . . . 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 3009 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ ((coeffβ€˜(𝐹 ∘f + 𝐺))β€˜π‘) β‰  0)
70 eqid 2733 . . . 4 (coeffβ€˜(𝐹 ∘f + 𝐺)) = (coeffβ€˜(𝐹 ∘f + 𝐺))
71 eqid 2733 . . . 4 (degβ€˜(𝐹 ∘f + 𝐺)) = (degβ€˜(𝐹 ∘f + 𝐺))
7270, 71dgrub 25748 . . 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 1372 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝑀 < 𝑁) β†’ 𝑁 ≀ (degβ€˜(𝐹 ∘f + 𝐺)))
745, 10letri3d 11356 . 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 712 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475  ifcif 4529   class class class wbr 5149  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∘f cof 7668  β„‚cc 11108  β„cr 11109  0cc0 11110   + caddc 11113   < clt 11248   ≀ cle 11249  β„•0cn0 12472  0𝑝c0p 25186  Polycply 25698  coeffccoe 25700  degcdgr 25701
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-sum 15633  df-0p 25187  df-ply 25702  df-coe 25704  df-dgr 25705
This theorem is referenced by:  dgrcolem2  25788  plyremlem  25817
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