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Theorem dgrsub2 35448
 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 deg
Assertion
Ref Expression
dgrsub2 Poly Poly deg coeff coeff deg

Proof of Theorem dgrsub2
StepHypRef Expression
1 simpr2 1004 . . 3 Poly Poly deg coeff coeff
2 dgr0 22951 . . . . 5 deg
3 nngt0 10605 . . . . 5
42, 3syl5eqbr 4428 . . . 4 deg
5 fveq2 5849 . . . . 5 deg deg
65breq1d 4405 . . . 4 deg deg
74, 6syl5ibrcom 222 . . 3 deg
81, 7syl 17 . 2 Poly Poly deg coeff coeff deg
9 plyssc 22889 . . . . . . . 8 Poly Poly
109sseli 3438 . . . . . . 7 Poly Poly
11 plyssc 22889 . . . . . . . 8 Poly Poly
1211sseli 3438 . . . . . . 7 Poly Poly
13 eqid 2402 . . . . . . . 8 deg deg
14 eqid 2402 . . . . . . . 8 deg deg
1513, 14dgrsub 22961 . . . . . . 7 Poly Poly deg deg deg deg deg
1610, 12, 15syl2an 475 . . . . . 6 Poly Poly deg deg deg deg deg
1716adantr 463 . . . . 5 Poly Poly deg coeff coeff deg deg deg deg deg
18 simpr1 1003 . . . . . . 7 Poly Poly deg coeff coeff deg
19 dgrsub2.a . . . . . . . . 9 deg
2019eqcomi 2415 . . . . . . . 8 deg
2120a1i 11 . . . . . . 7 Poly Poly deg coeff coeff deg
2218, 21ifeq12d 3905 . . . . . 6 Poly Poly deg coeff coeff deg deg deg deg deg deg
23 ifid 3922 . . . . . 6 deg deg
2422, 23syl6eq 2459 . . . . 5 Poly Poly deg coeff coeff deg deg deg deg
2517, 24breqtrd 4419 . . . 4 Poly Poly deg coeff coeff deg
26 eqid 2402 . . . . . . . . 9 coeff coeff
27 eqid 2402 . . . . . . . . 9 coeff coeff
2826, 27coesub 22946 . . . . . . . 8 Poly Poly coeff coeff coeff
2910, 12, 28syl2an 475 . . . . . . 7 Poly Poly coeff coeff coeff
3029adantr 463 . . . . . 6 Poly Poly deg coeff coeff coeff coeff coeff
3130fveq1d 5851 . . . . 5 Poly Poly deg coeff coeff coeff coeff coeff
321nnnn0d 10893 . . . . . 6 Poly Poly deg coeff coeff
3326coef3 22921 . . . . . . . . 9 Poly coeff
3433ad2antrr 724 . . . . . . . 8 Poly Poly deg coeff coeff coeff
35 ffn 5714 . . . . . . . 8 coeff coeff
3634, 35syl 17 . . . . . . 7 Poly Poly deg coeff coeff coeff
3727coef3 22921 . . . . . . . . 9 Poly coeff
3837ad2antlr 725 . . . . . . . 8 Poly Poly deg coeff coeff coeff
39 ffn 5714 . . . . . . . 8 coeff coeff
4038, 39syl 17 . . . . . . 7 Poly Poly deg coeff coeff coeff
41 nn0ex 10842 . . . . . . . 8
4241a1i 11 . . . . . . 7 Poly Poly deg coeff coeff
43 inidm 3648 . . . . . . 7
44 simplr3 1041 . . . . . . 7 Poly Poly deg coeff coeff coeff coeff
45 eqidd 2403 . . . . . . 7 Poly Poly deg coeff coeff coeff coeff
4636, 40, 42, 42, 43, 44, 45ofval 6530 . . . . . 6 Poly Poly deg coeff coeff coeff coeff coeff coeff
4732, 46mpdan 666 . . . . 5 Poly Poly deg coeff coeff coeff coeff coeff coeff
4838, 32ffvelrnd 6010 . . . . . 6 Poly Poly deg coeff coeff coeff
4948subidd 9955 . . . . 5 Poly Poly deg coeff coeff coeff coeff
5031, 47, 493eqtrd 2447 . . . 4 Poly Poly deg coeff coeff coeff
51 plysubcl 22911 . . . . . . 7 Poly Poly Poly
5210, 12, 51syl2an 475 . . . . . 6 Poly Poly Poly
5352adantr 463 . . . . 5 Poly Poly deg coeff coeff Poly
54 eqid 2402 . . . . . 6 deg deg
55 eqid 2402 . . . . . 6 coeff coeff
5654, 55dgrlt 22955 . . . . 5 Poly deg deg coeff
5753, 32, 56syl2anc 659 . . . 4 Poly Poly deg coeff coeff deg deg coeff
5825, 50, 57mpbir2and 923 . . 3 Poly Poly deg coeff coeff deg
5958ord 375 . 2 Poly Poly deg coeff coeff deg
608, 59pm2.61d 158 1 Poly Poly deg coeff coeff deg
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wo 366   wa 367   w3a 974   wceq 1405   wcel 1842  cvv 3059  cif 3885   class class class wbr 4395   wfn 5564  wf 5565  cfv 5569  (class class class)co 6278   cof 6519  cc 9520  cc0 9522   clt 9658   cle 9659   cmin 9841  cn 10576  cn0 10836  c0p 22368  Polycply 22873  coeffccoe 22875  degcdgr 22876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-0p 22369  df-ply 22877  df-coe 22879  df-dgr 22880 This theorem is referenced by:  mpaaeu  35463
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