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Theorem dgrsub2 27169
 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 20119 . . . . 5 deg
3 nngt0 9975 . . . . 5
42, 3syl5eqbr 4200 . . . 4 deg
5 fveq2 5682 . . . . 5 deg deg
65breq1d 4177 . . . 4 deg deg
74, 6syl5ibrcom 214 . . 3 deg
81, 7syl 16 . 2 Poly Poly deg coeff coeff deg
9 plyssc 20058 . . . . . . . 8 Poly Poly
109sseli 3301 . . . . . . 7 Poly Poly
11 plyssc 20058 . . . . . . . 8 Poly Poly
1211sseli 3301 . . . . . . 7 Poly Poly
13 eqid 2401 . . . . . . . 8 deg deg
14 eqid 2401 . . . . . . . 8 deg deg
1513, 14dgrsub 20129 . . . . . . 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 2405 . . . . . . . 8 deg
2120a1i 11 . . . . . . 7 Poly Poly deg coeff coeff deg
2218, 21ifeq12d 3712 . . . . . 6 Poly Poly deg coeff coeff deg deg deg deg deg deg
23 ifid 3728 . . . . . 6 deg deg
2422, 23syl6eq 2449 . . . . 5 Poly Poly deg coeff coeff deg deg deg deg
2517, 24breqtrd 4191 . . . 4 Poly Poly deg coeff coeff deg
26 eqid 2401 . . . . . . . . 9 coeff coeff
27 eqid 2401 . . . . . . . . 9 coeff coeff
2826, 27coesub 20114 . . . . . . . 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 5684 . . . . 5 Poly Poly deg coeff coeff coeff coeff coeff
321nnnn0d 10220 . . . . . 6 Poly Poly deg coeff coeff
3326coef3 20090 . . . . . . . . 9 Poly coeff
3433ad2antrr 707 . . . . . . . 8 Poly Poly deg coeff coeff coeff
35 ffn 5545 . . . . . . . 8 coeff coeff
3634, 35syl 16 . . . . . . 7 Poly Poly deg coeff coeff coeff
3727coef3 20090 . . . . . . . . 9 Poly coeff
3837ad2antlr 708 . . . . . . . 8 Poly Poly deg coeff coeff coeff
39 ffn 5545 . . . . . . . 8 coeff coeff
4038, 39syl 16 . . . . . . 7 Poly Poly deg coeff coeff coeff
41 nn0ex 10173 . . . . . . . 8
4241a1i 11 . . . . . . 7 Poly Poly deg coeff coeff
43 inidm 3507 . . . . . . 7
44 simplr3 1001 . . . . . . 7 Poly Poly deg coeff coeff coeff coeff
45 eqidd 2402 . . . . . . 7 Poly Poly deg coeff coeff coeff coeff
4636, 40, 42, 42, 43, 44, 45ofval 6267 . . . . . 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 5824 . . . . . 6 Poly Poly deg coeff coeff coeff
4948subidd 9345 . . . . 5 Poly Poly deg coeff coeff coeff coeff
5031, 47, 493eqtrd 2437 . . . 4 Poly Poly deg coeff coeff coeff
51 plysubcl 20080 . . . . . . 7 Poly Poly Poly
5210, 12, 51syl2an 464 . . . . . 6 Poly Poly Poly
5352adantr 452 . . . . 5 Poly Poly deg coeff coeff Poly
54 eqid 2401 . . . . . 6 deg deg
55 eqid 2401 . . . . . 6 coeff coeff
5654, 55dgrlt 20123 . . . . 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 1649   wcel 1721  cvv 2913  cif 3696   class class class wbr 4167   wfn 5403  wf 5404  cfv 5408  (class class class)co 6034   cof 6256  cc 8935  cc0 8937   clt 9067   cle 9068   cmin 9237  cn 9946  cn0 10167  c0p 19500  Polycply 20042  coeffccoe 20044  degcdgr 20045 This theorem is referenced by:  mpaaeu  27185 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2382  ax-rep 4275  ax-sep 4285  ax-nul 4293  ax-pow 4332  ax-pr 4358  ax-un 4655  ax-inf2 7543  ax-cnex 8993  ax-resscn 8994  ax-1cn 8995  ax-icn 8996  ax-addcl 8997  ax-addrcl 8998  ax-mulcl 8999  ax-mulrcl 9000  ax-mulcom 9001  ax-addass 9002  ax-mulass 9003  ax-distr 9004  ax-i2m1 9005  ax-1ne0 9006  ax-1rid 9007  ax-rnegex 9008  ax-rrecex 9009  ax-cnre 9010  ax-pre-lttri 9011  ax-pre-lttrn 9012  ax-pre-ltadd 9013  ax-pre-mulgt0 9014  ax-pre-sup 9015  ax-addf 9016 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2256  df-mo 2257  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2526  df-ne 2566  df-nel 2567  df-ral 2668  df-rex 2669  df-reu 2670  df-rmo 2671  df-rab 2672  df-v 2915  df-sbc 3119  df-csb 3209  df-dif 3280  df-un 3282  df-in 3284  df-ss 3291  df-pss 3293  df-nul 3586  df-if 3697  df-pw 3758  df-sn 3777  df-pr 3778  df-tp 3779  df-op 3780  df-uni 3972  df-int 4007  df-iun 4051  df-br 4168  df-opab 4222  df-mpt 4223  df-tr 4258  df-eprel 4449  df-id 4453  df-po 4458  df-so 4459  df-fr 4496  df-se 4497  df-we 4498  df-ord 4539  df-on 4540  df-lim 4541  df-suc 4542  df-om 4800  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5372  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-isom 5417  df-ov 6037  df-oprab 6038  df-mpt2 6039  df-of 6258  df-1st 6302  df-2nd 6303  df-riota 6499  df-recs 6583  df-rdg 6618  df-1o 6674  df-oadd 6678  df-er 6855  df-map 6970  df-pm 6971  df-en 7060  df-dom 7061  df-sdom 7062  df-fin 7063  df-sup 7395  df-oi 7426  df-card 7773  df-pnf 9069  df-mnf 9070  df-xr 9071  df-ltxr 9072  df-le 9073  df-sub 9239  df-neg 9240  df-div 9624  df-nn 9947  df-2 10004  df-3 10005  df-n0 10168  df-z 10229  df-uz 10435  df-rp 10559  df-fz 10990  df-fzo 11080  df-fl 11143  df-seq 11265  df-exp 11324  df-hash 11560  df-cj 11845  df-re 11846  df-im 11847  df-sqr 11981  df-abs 11982  df-clim 12223  df-rlim 12224  df-sum 12421  df-0p 19501  df-ply 20046  df-coe 20048  df-dgr 20049
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