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Theorem dgrsub2 27316
 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 964 . . 3 Poly Poly deg coeff coeff
2 dgr0 20180 . . . . 5 deg
3 nngt0 10029 . . . . 5
42, 3syl5eqbr 4245 . . . 4 deg
5 fveq2 5728 . . . . 5 deg deg
65breq1d 4222 . . . 4 deg deg
74, 6syl5ibrcom 214 . . 3 deg
81, 7syl 16 . 2 Poly Poly deg coeff coeff deg
9 plyssc 20119 . . . . . . . 8 Poly Poly
109sseli 3344 . . . . . . 7 Poly Poly
11 plyssc 20119 . . . . . . . 8 Poly Poly
1211sseli 3344 . . . . . . 7 Poly Poly
13 eqid 2436 . . . . . . . 8 deg deg
14 eqid 2436 . . . . . . . 8 deg deg
1513, 14dgrsub 20190 . . . . . . 7 Poly Poly deg deg deg deg deg
1610, 12, 15syl2an 464 . . . . . 6 Poly Poly deg deg deg deg deg
1716adantr 452 . . . . 5 Poly Poly deg coeff coeff deg deg deg deg deg
18 simpr1 963 . . . . . . 7 Poly Poly deg coeff coeff deg
19 dgrsub2.a . . . . . . . . 9 deg
2019eqcomi 2440 . . . . . . . 8 deg
2120a1i 11 . . . . . . 7 Poly Poly deg coeff coeff deg
2218, 21ifeq12d 3755 . . . . . 6 Poly Poly deg coeff coeff deg deg deg deg deg deg
23 ifid 3771 . . . . . 6 deg deg
2422, 23syl6eq 2484 . . . . 5 Poly Poly deg coeff coeff deg deg deg deg
2517, 24breqtrd 4236 . . . 4 Poly Poly deg coeff coeff deg
26 eqid 2436 . . . . . . . . 9 coeff coeff
27 eqid 2436 . . . . . . . . 9 coeff coeff
2826, 27coesub 20175 . . . . . . . 8 Poly Poly coeff coeff coeff
2910, 12, 28syl2an 464 . . . . . . 7 Poly Poly coeff coeff coeff
3029adantr 452 . . . . . 6 Poly Poly deg coeff coeff coeff coeff coeff
3130fveq1d 5730 . . . . 5 Poly Poly deg coeff coeff coeff coeff coeff
321nnnn0d 10274 . . . . . 6 Poly Poly deg coeff coeff
3326coef3 20151 . . . . . . . . 9 Poly coeff
3433ad2antrr 707 . . . . . . . 8 Poly Poly deg coeff coeff coeff
35 ffn 5591 . . . . . . . 8 coeff coeff
3634, 35syl 16 . . . . . . 7 Poly Poly deg coeff coeff coeff
3727coef3 20151 . . . . . . . . 9 Poly coeff
3837ad2antlr 708 . . . . . . . 8 Poly Poly deg coeff coeff coeff
39 ffn 5591 . . . . . . . 8 coeff coeff
4038, 39syl 16 . . . . . . 7 Poly Poly deg coeff coeff coeff
41 nn0ex 10227 . . . . . . . 8
4241a1i 11 . . . . . . 7 Poly Poly deg coeff coeff
43 inidm 3550 . . . . . . 7
44 simplr3 1001 . . . . . . 7 Poly Poly deg coeff coeff coeff coeff
45 eqidd 2437 . . . . . . 7 Poly Poly deg coeff coeff coeff coeff
4636, 40, 42, 42, 43, 44, 45ofval 6314 . . . . . 6 Poly Poly deg coeff coeff coeff coeff coeff coeff
4732, 46mpdan 650 . . . . 5 Poly Poly deg coeff coeff coeff coeff coeff coeff
4838, 32ffvelrnd 5871 . . . . . 6 Poly Poly deg coeff coeff coeff
4948subidd 9399 . . . . 5 Poly Poly deg coeff coeff coeff coeff
5031, 47, 493eqtrd 2472 . . . 4 Poly Poly deg coeff coeff coeff
51 plysubcl 20141 . . . . . . 7 Poly Poly Poly
5210, 12, 51syl2an 464 . . . . . 6 Poly Poly Poly
5352adantr 452 . . . . 5 Poly Poly deg coeff coeff Poly
54 eqid 2436 . . . . . 6 deg deg
55 eqid 2436 . . . . . 6 coeff coeff
5654, 55dgrlt 20184 . . . . 5 Poly deg deg coeff
5753, 32, 56syl2anc 643 . . . 4 Poly Poly deg coeff coeff deg deg coeff
5825, 50, 57mpbir2and 889 . . 3 Poly Poly deg coeff coeff deg
5958ord 367 . 2 Poly Poly deg coeff coeff deg
608, 59pm2.61d 152 1 Poly Poly deg coeff coeff deg
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wo 358   wa 359   w3a 936   wceq 1652   wcel 1725  cvv 2956  cif 3739   class class class wbr 4212   wfn 5449  wf 5450  cfv 5454  (class class class)co 6081   cof 6303  cc 8988  cc0 8990   clt 9120   cle 9121   cmin 9291  cn 10000  cn0 10221  c0p 19561  Polycply 20103  coeffccoe 20105  degcdgr 20106 This theorem is referenced by:  mpaaeu  27332 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110
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