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Theorem dgradd2 25429
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‘(𝐹f + 𝐺)) = 𝑁)

Proof of Theorem dgradd2
StepHypRef Expression
1 plyaddcl 25381 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f + 𝐺) ∈ (Poly‘ℂ))
213adant3 1131 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝐹f + 𝐺) ∈ (Poly‘ℂ))
3 dgrcl 25394 . . . . 5 ((𝐹f + 𝐺) ∈ (Poly‘ℂ) → (deg‘(𝐹f + 𝐺)) ∈ ℕ0)
42, 3syl 17 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) ∈ ℕ0)
54nn0red 12294 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) ∈ ℝ)
6 dgradd.2 . . . . . . 7 𝑁 = (deg‘𝐺)
7 dgrcl 25394 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
86, 7eqeltrid 2843 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
983ad2ant2 1133 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℕ0)
109nn0red 12294 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℝ)
11 dgradd.1 . . . . . . 7 𝑀 = (deg‘𝐹)
12 dgrcl 25394 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
1311, 12eqeltrid 2843 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
14133ad2ant1 1132 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℕ0)
1514nn0red 12294 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℝ)
1610, 15ifcld 4505 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
1711, 6dgradd 25428 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
18173adant3 1131 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
1910leidd 11541 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁𝑁)
20 simp3 1137 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 < 𝑁)
2115, 10, 20ltled 11123 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀𝑁)
22 breq1 5077 . . . . 5 (𝑁 = if(𝑀𝑁, 𝑁, 𝑀) → (𝑁𝑁 ↔ if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁))
23 breq1 5077 . . . . 5 (𝑀 = if(𝑀𝑁, 𝑁, 𝑀) → (𝑀𝑁 ↔ if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁))
2422, 23ifboth 4498 . . . 4 ((𝑁𝑁𝑀𝑁) → if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁)
2519, 21, 24syl2anc 584 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁)
265, 16, 10, 18, 25letrd 11132 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) ≤ 𝑁)
27 eqid 2738 . . . . . . . 8 (coeff‘𝐹) = (coeff‘𝐹)
28 eqid 2738 . . . . . . . 8 (coeff‘𝐺) = (coeff‘𝐺)
2927, 28coeadd 25412 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f + 𝐺)) = ((coeff‘𝐹) ∘f + (coeff‘𝐺)))
30293adant3 1131 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘(𝐹f + 𝐺)) = ((coeff‘𝐹) ∘f + (coeff‘𝐺)))
3130fveq1d 6776 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹f + 𝐺))‘𝑁) = (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁))
3227coef3 25393 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
33323ad2ant1 1132 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹):ℕ0⟶ℂ)
3433ffnd 6601 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹) Fn ℕ0)
3528coef3 25393 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
36353ad2ant2 1133 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺):ℕ0⟶ℂ)
3736ffnd 6601 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺) Fn ℕ0)
38 nn0ex 12239 . . . . . . . 8 0 ∈ V
3938a1i 11 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ℕ0 ∈ V)
40 inidm 4152 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
4115, 10ltnled 11122 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝑀 < 𝑁 ↔ ¬ 𝑁𝑀))
4220, 41mpbid 231 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ¬ 𝑁𝑀)
43 simp1 1135 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐹 ∈ (Poly‘𝑆))
4427, 11dgrub 25395 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝐹)‘𝑁) ≠ 0) → 𝑁𝑀)
45443expia 1120 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹)‘𝑁) ≠ 0 → 𝑁𝑀))
4643, 9, 45syl2anc 584 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹)‘𝑁) ≠ 0 → 𝑁𝑀))
4746necon1bd 2961 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (¬ 𝑁𝑀 → ((coeff‘𝐹)‘𝑁) = 0))
4842, 47mpd 15 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐹)‘𝑁) = 0)
4948adantr 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))
5134, 37, 39, 39, 40, 49, 50ofval 7544 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
529, 51mpdan 684 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
5336, 9ffvelrnd 6962 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
5453addid2d 11176 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (0 + ((coeff‘𝐺)‘𝑁)) = ((coeff‘𝐺)‘𝑁))
5531, 52, 543eqtrd 2782 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹f + 𝐺))‘𝑁) = ((coeff‘𝐺)‘𝑁))
56 simp2 1136 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐺 ∈ (Poly‘𝑆))
57 0red 10978 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ∈ ℝ)
5814nn0ge0d 12296 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ≤ 𝑀)
5957, 15, 10, 58, 20lelttrd 11133 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 < 𝑁)
6059gt0ne0d 11539 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≠ 0)
616, 28dgreq0 25426 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘𝑁) = 0))
62 fveq2 6774 . . . . . . . 8 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
63 dgr0 25423 . . . . . . . . 9 (deg‘0𝑝) = 0
6463eqcomi 2747 . . . . . . . 8 0 = (deg‘0𝑝)
6562, 6, 643eqtr4g 2803 . . . . . . 7 (𝐺 = 0𝑝𝑁 = 0)
6661, 65syl6bir 253 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (((coeff‘𝐺)‘𝑁) = 0 → 𝑁 = 0))
6766necon3d 2964 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (𝑁 ≠ 0 → ((coeff‘𝐺)‘𝑁) ≠ 0))
6856, 60, 67sylc 65 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ≠ 0)
6955, 68eqnetrd 3011 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹f + 𝐺))‘𝑁) ≠ 0)
70 eqid 2738 . . . 4 (coeff‘(𝐹f + 𝐺)) = (coeff‘(𝐹f + 𝐺))
71 eqid 2738 . . . 4 (deg‘(𝐹f + 𝐺)) = (deg‘(𝐹f + 𝐺))
7270, 71dgrub 25395 . . 3 (((𝐹f + 𝐺) ∈ (Poly‘ℂ) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘(𝐹f + 𝐺))‘𝑁) ≠ 0) → 𝑁 ≤ (deg‘(𝐹f + 𝐺)))
732, 9, 69, 72syl3anc 1370 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≤ (deg‘(𝐹f + 𝐺)))
745, 10letri3d 11117 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((deg‘(𝐹f + 𝐺)) = 𝑁 ↔ ((deg‘(𝐹f + 𝐺)) ≤ 𝑁𝑁 ≤ (deg‘(𝐹f + 𝐺)))))
7526, 73, 74mpbir2and 710 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  Vcvv 3432  ifcif 4459   class class class wbr 5074  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531  cc 10869  cr 10870  0cc0 10871   + caddc 10874   < clt 11009  cle 11010  0cn0 12233  0𝑝c0p 24833  Polycply 25345  coeffccoe 25347  degcdgr 25348
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 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
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-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  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 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-0p 24834  df-ply 25349  df-coe 25351  df-dgr 25352
This theorem is referenced by:  dgrcolem2  25435  plyremlem  25464
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