Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dgrsub2 Structured version   Visualization version   GIF version

Theorem dgrsub2 38226
Description: Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Hypothesis
Ref Expression
dgrsub2.a 𝑁 = (deg‘𝐹)
Assertion
Ref Expression
dgrsub2 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹𝑓𝐺)) < 𝑁)

Proof of Theorem dgrsub2
StepHypRef Expression
1 simpr2 1236 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ)
2 dgr0 24238 . . . . 5 (deg‘0𝑝) = 0
3 nngt0 11262 . . . . 5 (𝑁 ∈ ℕ → 0 < 𝑁)
42, 3syl5eqbr 4840 . . . 4 (𝑁 ∈ ℕ → (deg‘0𝑝) < 𝑁)
5 fveq2 6354 . . . . 5 ((𝐹𝑓𝐺) = 0𝑝 → (deg‘(𝐹𝑓𝐺)) = (deg‘0𝑝))
65breq1d 4815 . . . 4 ((𝐹𝑓𝐺) = 0𝑝 → ((deg‘(𝐹𝑓𝐺)) < 𝑁 ↔ (deg‘0𝑝) < 𝑁))
74, 6syl5ibrcom 237 . . 3 (𝑁 ∈ ℕ → ((𝐹𝑓𝐺) = 0𝑝 → (deg‘(𝐹𝑓𝐺)) < 𝑁))
81, 7syl 17 . 2 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((𝐹𝑓𝐺) = 0𝑝 → (deg‘(𝐹𝑓𝐺)) < 𝑁))
9 plyssc 24176 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
109sseli 3741 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
11 plyssc 24176 . . . . . . . 8 (Poly‘𝑇) ⊆ (Poly‘ℂ)
1211sseli 3741 . . . . . . 7 (𝐺 ∈ (Poly‘𝑇) → 𝐺 ∈ (Poly‘ℂ))
13 eqid 2761 . . . . . . . 8 (deg‘𝐹) = (deg‘𝐹)
14 eqid 2761 . . . . . . . 8 (deg‘𝐺) = (deg‘𝐺)
1513, 14dgrsub 24248 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (deg‘(𝐹𝑓𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
1610, 12, 15syl2an 495 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (deg‘(𝐹𝑓𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
1716adantr 472 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹𝑓𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
18 simpr1 1234 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘𝐺) = 𝑁)
19 dgrsub2.a . . . . . . . . 9 𝑁 = (deg‘𝐹)
2019eqcomi 2770 . . . . . . . 8 (deg‘𝐹) = 𝑁
2120a1i 11 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘𝐹) = 𝑁)
2218, 21ifeq12d 4251 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)) = if((deg‘𝐹) ≤ (deg‘𝐺), 𝑁, 𝑁))
23 ifid 4270 . . . . . 6 if((deg‘𝐹) ≤ (deg‘𝐺), 𝑁, 𝑁) = 𝑁
2422, 23syl6eq 2811 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)) = 𝑁)
2517, 24breqtrd 4831 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹𝑓𝐺)) ≤ 𝑁)
26 eqid 2761 . . . . . . . . 9 (coeff‘𝐹) = (coeff‘𝐹)
27 eqid 2761 . . . . . . . . 9 (coeff‘𝐺) = (coeff‘𝐺)
2826, 27coesub 24233 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘(𝐹𝑓𝐺)) = ((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺)))
2910, 12, 28syl2an 495 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (coeff‘(𝐹𝑓𝐺)) = ((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺)))
3029adantr 472 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘(𝐹𝑓𝐺)) = ((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺)))
3130fveq1d 6356 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘(𝐹𝑓𝐺))‘𝑁) = (((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺))‘𝑁))
321nnnn0d 11564 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ0)
3326coef3 24208 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
3433ad2antrr 764 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐹):ℕ0⟶ℂ)
35 ffn 6207 . . . . . . . 8 ((coeff‘𝐹):ℕ0⟶ℂ → (coeff‘𝐹) Fn ℕ0)
3634, 35syl 17 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐹) Fn ℕ0)
3727coef3 24208 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑇) → (coeff‘𝐺):ℕ0⟶ℂ)
3837ad2antlr 765 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐺):ℕ0⟶ℂ)
39 ffn 6207 . . . . . . . 8 ((coeff‘𝐺):ℕ0⟶ℂ → (coeff‘𝐺) Fn ℕ0)
4038, 39syl 17 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐺) Fn ℕ0)
41 nn0ex 11511 . . . . . . . 8 0 ∈ V
4241a1i 11 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ℕ0 ∈ V)
43 inidm 3966 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
44 simplr3 1265 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))
45 eqidd 2762 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐺)‘𝑁) = ((coeff‘𝐺)‘𝑁))
4636, 40, 42, 42, 43, 44, 45ofval 7073 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) ∧ 𝑁 ∈ ℕ0) → (((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺))‘𝑁) = (((coeff‘𝐺)‘𝑁) − ((coeff‘𝐺)‘𝑁)))
4732, 46mpdan 705 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺))‘𝑁) = (((coeff‘𝐺)‘𝑁) − ((coeff‘𝐺)‘𝑁)))
4838, 32ffvelrnd 6525 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
4948subidd 10593 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (((coeff‘𝐺)‘𝑁) − ((coeff‘𝐺)‘𝑁)) = 0)
5031, 47, 493eqtrd 2799 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘(𝐹𝑓𝐺))‘𝑁) = 0)
51 plysubcl 24198 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
5210, 12, 51syl2an 495 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
5352adantr 472 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
54 eqid 2761 . . . . . 6 (deg‘(𝐹𝑓𝐺)) = (deg‘(𝐹𝑓𝐺))
55 eqid 2761 . . . . . 6 (coeff‘(𝐹𝑓𝐺)) = (coeff‘(𝐹𝑓𝐺))
5654, 55dgrlt 24242 . . . . 5 (((𝐹𝑓𝐺) ∈ (Poly‘ℂ) ∧ 𝑁 ∈ ℕ0) → (((𝐹𝑓𝐺) = 0𝑝 ∨ (deg‘(𝐹𝑓𝐺)) < 𝑁) ↔ ((deg‘(𝐹𝑓𝐺)) ≤ 𝑁 ∧ ((coeff‘(𝐹𝑓𝐺))‘𝑁) = 0)))
5753, 32, 56syl2anc 696 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (((𝐹𝑓𝐺) = 0𝑝 ∨ (deg‘(𝐹𝑓𝐺)) < 𝑁) ↔ ((deg‘(𝐹𝑓𝐺)) ≤ 𝑁 ∧ ((coeff‘(𝐹𝑓𝐺))‘𝑁) = 0)))
5825, 50, 57mpbir2and 995 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((𝐹𝑓𝐺) = 0𝑝 ∨ (deg‘(𝐹𝑓𝐺)) < 𝑁))
5958ord 391 . 2 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (¬ (𝐹𝑓𝐺) = 0𝑝 → (deg‘(𝐹𝑓𝐺)) < 𝑁))
608, 59pm2.61d 170 1 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹𝑓𝐺)) < 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2140  Vcvv 3341  ifcif 4231   class class class wbr 4805   Fn wfn 6045  wf 6046  cfv 6050  (class class class)co 6815  𝑓 cof 7062  cc 10147  0cc0 10149   < clt 10287  cle 10288  cmin 10479  cn 11233  0cn0 11505  0𝑝c0p 23656  Polycply 24160  coeffccoe 24162  degcdgr 24163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-inf2 8714  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-pre-sup 10227  ax-addf 10228
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-se 5227  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-isom 6059  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-of 7064  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-oadd 7735  df-er 7914  df-map 8028  df-pm 8029  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-sup 8516  df-inf 8517  df-oi 8583  df-card 8976  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-div 10898  df-nn 11234  df-2 11292  df-3 11293  df-n0 11506  df-z 11591  df-uz 11901  df-rp 12047  df-fz 12541  df-fzo 12681  df-fl 12808  df-seq 13017  df-exp 13076  df-hash 13333  df-cj 14059  df-re 14060  df-im 14061  df-sqrt 14195  df-abs 14196  df-clim 14439  df-rlim 14440  df-sum 14637  df-0p 23657  df-ply 24164  df-coe 24166  df-dgr 24167
This theorem is referenced by:  mpaaeu  38241
  Copyright terms: Public domain W3C validator