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Theorem dgrsub2 36724
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 1061 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ)
2 dgr0 23822 . . . . 5 (deg‘0𝑝) = 0
3 nngt0 10926 . . . . 5 (𝑁 ∈ ℕ → 0 < 𝑁)
42, 3syl5eqbr 4618 . . . 4 (𝑁 ∈ ℕ → (deg‘0𝑝) < 𝑁)
5 fveq2 6103 . . . . 5 ((𝐹𝑓𝐺) = 0𝑝 → (deg‘(𝐹𝑓𝐺)) = (deg‘0𝑝))
65breq1d 4593 . . . 4 ((𝐹𝑓𝐺) = 0𝑝 → ((deg‘(𝐹𝑓𝐺)) < 𝑁 ↔ (deg‘0𝑝) < 𝑁))
74, 6syl5ibrcom 236 . . 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 23760 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
109sseli 3564 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
11 plyssc 23760 . . . . . . . 8 (Poly‘𝑇) ⊆ (Poly‘ℂ)
1211sseli 3564 . . . . . . 7 (𝐺 ∈ (Poly‘𝑇) → 𝐺 ∈ (Poly‘ℂ))
13 eqid 2610 . . . . . . . 8 (deg‘𝐹) = (deg‘𝐹)
14 eqid 2610 . . . . . . . 8 (deg‘𝐺) = (deg‘𝐺)
1513, 14dgrsub 23832 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (deg‘(𝐹𝑓𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
1610, 12, 15syl2an 493 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (deg‘(𝐹𝑓𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
1716adantr 480 . . . . 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 1060 . . . . . . 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 2619 . . . . . . . 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 4056 . . . . . 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 4075 . . . . . 6 if((deg‘𝐹) ≤ (deg‘𝐺), 𝑁, 𝑁) = 𝑁
2422, 23syl6eq 2660 . . . . 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 4609 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹𝑓𝐺)) ≤ 𝑁)
26 eqid 2610 . . . . . . . . 9 (coeff‘𝐹) = (coeff‘𝐹)
27 eqid 2610 . . . . . . . . 9 (coeff‘𝐺) = (coeff‘𝐺)
2826, 27coesub 23817 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘(𝐹𝑓𝐺)) = ((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺)))
2910, 12, 28syl2an 493 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (coeff‘(𝐹𝑓𝐺)) = ((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺)))
3029adantr 480 . . . . . 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 6105 . . . . 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 11228 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ0)
3326coef3 23792 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
3433ad2antrr 758 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐹):ℕ0⟶ℂ)
35 ffn 5958 . . . . . . . 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 23792 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑇) → (coeff‘𝐺):ℕ0⟶ℂ)
3837ad2antlr 759 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐺):ℕ0⟶ℂ)
39 ffn 5958 . . . . . . . 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 11175 . . . . . . . 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 3784 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
44 simplr3 1098 . . . . . . 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 2611 . . . . . . 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 6804 . . . . . 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 699 . . . . 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 6268 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
4948subidd 10259 . . . . 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 2648 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘(𝐹𝑓𝐺))‘𝑁) = 0)
51 plysubcl 23782 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
5210, 12, 51syl2an 493 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
5352adantr 480 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
54 eqid 2610 . . . . . 6 (deg‘(𝐹𝑓𝐺)) = (deg‘(𝐹𝑓𝐺))
55 eqid 2610 . . . . . 6 (coeff‘(𝐹𝑓𝐺)) = (coeff‘(𝐹𝑓𝐺))
5654, 55dgrlt 23826 . . . . 5 (((𝐹𝑓𝐺) ∈ (Poly‘ℂ) ∧ 𝑁 ∈ ℕ0) → (((𝐹𝑓𝐺) = 0𝑝 ∨ (deg‘(𝐹𝑓𝐺)) < 𝑁) ↔ ((deg‘(𝐹𝑓𝐺)) ≤ 𝑁 ∧ ((coeff‘(𝐹𝑓𝐺))‘𝑁) = 0)))
5753, 32, 56syl2anc 691 . . . 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 959 . . 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 169 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 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  ifcif 4036   class class class wbr 4583   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  𝑓 cof 6793  cc 9813  0cc0 9815   < clt 9953  cle 9954  cmin 10145  cn 10897  0cn0 11169  0𝑝c0p 23242  Polycply 23744  coeffccoe 23746  degcdgr 23747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750  df-dgr 23751
This theorem is referenced by:  mpaaeu  36739
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