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Theorem dgradd2 26323
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 26274 . . . . . 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 26287 . . . . 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 12586 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) ∈ ℝ)
6 dgradd.2 . . . . . . 7 𝑁 = (deg‘𝐺)
7 dgrcl 26287 . . . . . . 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 12586 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ∈ ℝ)
11 dgradd.1 . . . . . . 7 𝑀 = (deg‘𝐹)
12 dgrcl 26287 . . . . . . 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 12586 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀 ∈ ℝ)
1610, 15ifcld 4577 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
1711, 6dgradd 26322 . . . 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 11827 . . . 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 11407 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑀𝑁)
22 breq1 5151 . . . . 5 (𝑁 = if(𝑀𝑁, 𝑁, 𝑀) → (𝑁𝑁 ↔ if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁))
23 breq1 5151 . . . . 5 (𝑀 = if(𝑀𝑁, 𝑁, 𝑀) → (𝑀𝑁 ↔ if(𝑀𝑁, 𝑁, 𝑀) ≤ 𝑁))
2422, 23ifboth 4570 . . . 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 11416 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) ≤ 𝑁)
27 eqid 2735 . . . . . . . 8 (coeff‘𝐹) = (coeff‘𝐹)
28 eqid 2735 . . . . . . . 8 (coeff‘𝐺) = (coeff‘𝐺)
2927, 28coeadd 26305 . . . . . . 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 6909 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹f + 𝐺))‘𝑁) = (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁))
3227coef3 26286 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
33323ad2ant1 1132 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹):ℕ0⟶ℂ)
3433ffnd 6738 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐹) Fn ℕ0)
3528coef3 26286 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
36353ad2ant2 1133 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺):ℕ0⟶ℂ)
3736ffnd 6738 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (coeff‘𝐺) Fn ℕ0)
38 nn0ex 12530 . . . . . . . 8 0 ∈ V
3938a1i 11 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ℕ0 ∈ V)
40 inidm 4235 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
4115, 10ltnled 11406 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (𝑀 < 𝑁 ↔ ¬ 𝑁𝑀))
4220, 41mpbid 232 . . . . . . . . 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 26288 . . . . . . . . . . . 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 2956 . . . . . . . . 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 480 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹)‘𝑁) = 0)
50 eqidd 2736 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐺)‘𝑁) = ((coeff‘𝐺)‘𝑁))
5134, 37, 39, 39, 40, 49, 50ofval 7708 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
529, 51mpdan 687 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (((coeff‘𝐹) ∘f + (coeff‘𝐺))‘𝑁) = (0 + ((coeff‘𝐺)‘𝑁)))
5336, 9ffvelcdmd 7105 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
5453addlidd 11460 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (0 + ((coeff‘𝐺)‘𝑁)) = ((coeff‘𝐺)‘𝑁))
5531, 52, 543eqtrd 2779 . . . 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 11262 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ∈ ℝ)
5814nn0ge0d 12588 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 ≤ 𝑀)
5957, 15, 10, 58, 20lelttrd 11417 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 0 < 𝑁)
6059gt0ne0d 11825 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → 𝑁 ≠ 0)
616, 28dgreq0 26320 . . . . . . 7 (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘𝑁) = 0))
62 fveq2 6907 . . . . . . . 8 (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝))
63 dgr0 26317 . . . . . . . . 9 (deg‘0𝑝) = 0
6463eqcomi 2744 . . . . . . . 8 0 = (deg‘0𝑝)
6562, 6, 643eqtr4g 2800 . . . . . . 7 (𝐺 = 0𝑝𝑁 = 0)
6661, 65biimtrrdi 254 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (((coeff‘𝐺)‘𝑁) = 0 → 𝑁 = 0))
6766necon3d 2959 . . . . 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 3006 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((coeff‘(𝐹f + 𝐺))‘𝑁) ≠ 0)
70 eqid 2735 . . . 4 (coeff‘(𝐹f + 𝐺)) = (coeff‘(𝐹f + 𝐺))
71 eqid 2735 . . . 4 (deg‘(𝐹f + 𝐺)) = (deg‘(𝐹f + 𝐺))
7270, 71dgrub 26288 . . 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 11401 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → ((deg‘(𝐹f + 𝐺)) = 𝑁 ↔ ((deg‘(𝐹f + 𝐺)) ≤ 𝑁𝑁 ≤ (deg‘(𝐹f + 𝐺)))))
7526, 73, 74mpbir2and 713 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 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  ifcif 4531   class class class wbr 5148  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  cc 11151  cr 11152  0cc0 11153   + caddc 11156   < clt 11293  cle 11294  0cn0 12524  0𝑝c0p 25718  Polycply 26238  coeffccoe 26240  degcdgr 26241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-0p 25719  df-ply 26242  df-coe 26244  df-dgr 26245
This theorem is referenced by:  dgrcolem2  26329  plyremlem  26361
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