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Theorem dgradd2 24069
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 24021 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + 𝐺) ∈ (Poly‘ℂ))
213adant3 1101 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝐹𝑓 + 𝐺) ∈ (Poly‘ℂ))
3 dgrcl 24034 . . . . 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 11390 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹𝑓 + 𝐺)) ∈ ℝ)
6 dgradd.2 . . . . . . 7 𝑁 = (deg‘𝐺)
7 dgrcl 24034 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
86, 7syl5eqel 2734 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
983ad2ant2 1103 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℕ0)
109nn0red 11390 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℝ)
11 dgradd.1 . . . . . . 7 𝑀 = (deg‘𝐹)
12 dgrcl 24034 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
1311, 12syl5eqel 2734 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
14133ad2ant1 1102 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℕ0)
1514nn0red 11390 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℝ)
1610, 15ifcld 4164 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
1711, 6dgradd 24068 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
18173adant3 1101 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
1910leidd 10632 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁𝑁)
20 simp3 1083 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 < 𝑁)
2115, 10, 20ltled 10223 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀𝑁)
22 breq1 4688 . . . . 5 (𝑁 = if(𝑀𝑁, 𝑁, 𝑀) → (𝑁𝑁 ↔ if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁))
23 breq1 4688 . . . . 5 (𝑀 = if(𝑀𝑁, 𝑁, 𝑀) → (𝑀𝑁 ↔ if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁))
2422, 23ifboth 4157 . . . 4 ((𝑁𝑁𝑀𝑁) → if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁)
2519, 21, 24syl2anc 694 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁)
265, 16, 10, 18, 25letrd 10232 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹𝑓 + 𝐺)) ≤ 𝑁)
27 eqid 2651 . . . . . . . 8 (coeff‘𝐹) = (coeff‘𝐹)
28 eqid 2651 . . . . . . . 8 (coeff‘𝐺) = (coeff‘𝐺)
2927, 28coeadd 24052 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + 𝐺)) = ((coeff‘𝐹) ∘𝑓 + (coeff‘𝐺)))
30293adant3 1101 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘(𝐹𝑓 + 𝐺)) = ((coeff‘𝐹) ∘𝑓 + (coeff‘𝐺)))
3130fveq1d 6231 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹𝑓 + 𝐺))‘𝑁) = (((coeff‘𝐹) ∘𝑓 + (coeff‘𝐺))‘𝑁))
3227coef3 24033 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
33323ad2ant1 1102 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹):ℕ0⟶ℂ)
34 ffn 6083 . . . . . . . 8 ((coeff‘𝐹):ℕ0⟶ℂ → (coeff‘𝐹) Fn ℕ0)
3533, 34syl 17 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹) Fn ℕ0)
3628coef3 24033 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
37363ad2ant2 1103 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺):ℕ0⟶ℂ)
38 ffn 6083 . . . . . . . 8 ((coeff‘𝐺):ℕ0⟶ℂ → (coeff‘𝐺) Fn ℕ0)
3937, 38syl 17 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺) Fn ℕ0)
40 nn0ex 11336 . . . . . . . 8 0 ∈ V
4140a1i 11 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ℕ0 ∈ V)
42 inidm 3855 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
4315, 10ltnled 10222 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝑀 < 𝑁 ↔ ¬ 𝑁𝑀))
4420, 43mpbid 222 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ¬ 𝑁𝑀)
45 simp1 1081 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐹 ∈ (Poly‘𝑆))
4627, 11dgrub 24035 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝐹)‘𝑁) ≠ 0) → 𝑁𝑀)
47463expia 1286 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹)‘𝑁) ≠ 0 → 𝑁𝑀))
4845, 9, 47syl2anc 694 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹)‘𝑁) ≠ 0 → 𝑁𝑀))
4948necon1bd 2841 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (¬ 𝑁𝑀 → ((coeff‘𝐹)‘𝑁) = 0))
5044, 49mpd 15 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐹)‘𝑁) = 0)
5150adantr 480 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹)‘𝑁) = 0)
52 eqidd 2652 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐺)‘𝑁) = ((coeff‘𝐺)‘𝑁))
5335, 39, 41, 41, 42, 51, 52ofval 6948 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹) ∘𝑓 + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
549, 53mpdan 703 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹) ∘𝑓 + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
5537, 9ffvelrnd 6400 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
5655addid2d 10275 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (0 + ((coeff‘𝐺)‘𝑁)) = ((coeff‘𝐺)‘𝑁))
5731, 54, 563eqtrd 2689 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹𝑓 + 𝐺))‘𝑁) = ((coeff‘𝐺)‘𝑁))
58 simp2 1082 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝐺 ∈ (Poly‘𝑆))
59 0red 10079 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ∈ ℝ)
6014nn0ge0d 11392 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ≤ 𝑀)
6159, 15, 10, 60, 20lelttrd 10233 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 < 𝑁)
6261gt0ne0d 10630 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≠ 0)
636, 28dgreq0 24066 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘𝑁) = 0))
64 fveq2 6229 . . . . . . . 8 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
65 dgr0 24063 . . . . . . . . 9 (deg‘0𝑝) = 0
6665eqcomi 2660 . . . . . . . 8 0 = (deg‘0𝑝)
6764, 6, 663eqtr4g 2710 . . . . . . 7 (𝐺 = 0𝑝𝑁 = 0)
6863, 67syl6bir 244 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (((coeff‘𝐺)‘𝑁) = 0 → 𝑁 = 0))
6968necon3d 2844 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (𝑁 ≠ 0 → ((coeff‘𝐺)‘𝑁) ≠ 0))
7058, 62, 69sylc 65 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ≠ 0)
7157, 70eqnetrd 2890 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹𝑓 + 𝐺))‘𝑁) ≠ 0)
72 eqid 2651 . . . 4 (coeff‘(𝐹𝑓 + 𝐺)) = (coeff‘(𝐹𝑓 + 𝐺))
73 eqid 2651 . . . 4 (deg‘(𝐹𝑓 + 𝐺)) = (deg‘(𝐹𝑓 + 𝐺))
7472, 73dgrub 24035 . . 3 (((𝐹𝑓 + 𝐺) ∈ (Poly‘ℂ) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘(𝐹𝑓 + 𝐺))‘𝑁) ≠ 0) → 𝑁 ≤ (deg‘(𝐹𝑓 + 𝐺)))
752, 9, 71, 74syl3anc 1366 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≤ (deg‘(𝐹𝑓 + 𝐺)))
765, 10letri3d 10217 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((deg‘(𝐹𝑓 + 𝐺)) = 𝑁 ↔ ((deg‘(𝐹𝑓 + 𝐺)) ≤ 𝑁𝑁 ≤ (deg‘(𝐹𝑓 + 𝐺)))))
7726, 75, 76mpbir2and 977 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  ifcif 4119   class class class wbr 4685   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  cc 9972  cr 9973  0cc0 9974   + caddc 9977   < clt 10112  cle 10113  0cn0 11330  0𝑝c0p 23481  Polycply 23985  coeffccoe 23987  degcdgr 23988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-0p 23482  df-ply 23989  df-coe 23991  df-dgr 23992
This theorem is referenced by:  dgrcolem2  24075  plyremlem  24104
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