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Theorem dgrsub2 37183
 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 1066 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ)
2 dgr0 23922 . . . . 5 (deg‘0𝑝) = 0
3 nngt0 10993 . . . . 5 (𝑁 ∈ ℕ → 0 < 𝑁)
42, 3syl5eqbr 4648 . . . 4 (𝑁 ∈ ℕ → (deg‘0𝑝) < 𝑁)
5 fveq2 6148 . . . . 5 ((𝐹𝑓𝐺) = 0𝑝 → (deg‘(𝐹𝑓𝐺)) = (deg‘0𝑝))
65breq1d 4623 . . . 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 23860 . . . . . . . 8 (Poly‘𝑆) ⊆ (Poly‘ℂ)
109sseli 3579 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
11 plyssc 23860 . . . . . . . 8 (Poly‘𝑇) ⊆ (Poly‘ℂ)
1211sseli 3579 . . . . . . 7 (𝐺 ∈ (Poly‘𝑇) → 𝐺 ∈ (Poly‘ℂ))
13 eqid 2621 . . . . . . . 8 (deg‘𝐹) = (deg‘𝐹)
14 eqid 2621 . . . . . . . 8 (deg‘𝐺) = (deg‘𝐺)
1513, 14dgrsub 23932 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (deg‘(𝐹𝑓𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
1610, 12, 15syl2an 494 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (deg‘(𝐹𝑓𝐺)) ≤ if((deg‘𝐹) ≤ (deg‘𝐺), (deg‘𝐺), (deg‘𝐹)))
1716adantr 481 . . . . 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 1065 . . . . . . 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 2630 . . . . . . . 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 4078 . . . . . 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 4097 . . . . . 6 if((deg‘𝐹) ≤ (deg‘𝐺), 𝑁, 𝑁) = 𝑁
2422, 23syl6eq 2671 . . . . 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 4639 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹𝑓𝐺)) ≤ 𝑁)
26 eqid 2621 . . . . . . . . 9 (coeff‘𝐹) = (coeff‘𝐹)
27 eqid 2621 . . . . . . . . 9 (coeff‘𝐺) = (coeff‘𝐺)
2826, 27coesub 23917 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘(𝐹𝑓𝐺)) = ((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺)))
2910, 12, 28syl2an 494 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (coeff‘(𝐹𝑓𝐺)) = ((coeff‘𝐹) ∘𝑓 − (coeff‘𝐺)))
3029adantr 481 . . . . . 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 6150 . . . . 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 11295 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → 𝑁 ∈ ℕ0)
3326coef3 23892 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
3433ad2antrr 761 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐹):ℕ0⟶ℂ)
35 ffn 6002 . . . . . . . 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 23892 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑇) → (coeff‘𝐺):ℕ0⟶ℂ)
3837ad2antlr 762 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (coeff‘𝐺):ℕ0⟶ℂ)
39 ffn 6002 . . . . . . . 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 11242 . . . . . . . 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 3800 . . . . . . 7 (ℕ0 ∩ ℕ0) = ℕ0
44 simplr3 1103 . . . . . . 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 2622 . . . . . . 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 6859 . . . . . 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 701 . . . . 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 6316 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘𝐺)‘𝑁) ∈ ℂ)
4948subidd 10324 . . . . 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 2659 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((coeff‘(𝐹𝑓𝐺))‘𝑁) = 0)
51 plysubcl 23882 . . . . . . 7 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
5210, 12, 51syl2an 494 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
5352adantr 481 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))
54 eqid 2621 . . . . . 6 (deg‘(𝐹𝑓𝐺)) = (deg‘(𝐹𝑓𝐺))
55 eqid 2621 . . . . . 6 (coeff‘(𝐹𝑓𝐺)) = (coeff‘(𝐹𝑓𝐺))
5654, 55dgrlt 23926 . . . . 5 (((𝐹𝑓𝐺) ∈ (Poly‘ℂ) ∧ 𝑁 ∈ ℕ0) → (((𝐹𝑓𝐺) = 0𝑝 ∨ (deg‘(𝐹𝑓𝐺)) < 𝑁) ↔ ((deg‘(𝐹𝑓𝐺)) ≤ 𝑁 ∧ ((coeff‘(𝐹𝑓𝐺))‘𝑁) = 0)))
5753, 32, 56syl2anc 692 . . . 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 956 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → ((𝐹𝑓𝐺) = 0𝑝 ∨ (deg‘(𝐹𝑓𝐺)) < 𝑁))
5958ord 392 . 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 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ifcif 4058   class class class wbr 4613   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ∘𝑓 cof 6848  ℂcc 9878  0cc0 9880   < clt 10018   ≤ cle 10019   − cmin 10210  ℕcn 10964  ℕ0cn0 11236  0𝑝c0p 23342  Polycply 23844  coeffccoe 23846  degcdgr 23847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-0p 23343  df-ply 23848  df-coe 23850  df-dgr 23851 This theorem is referenced by:  mpaaeu  37198
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