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Theorem elmnc 27318
 Description: Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Assertion
Ref Expression
elmnc Poly coeffdeg

Proof of Theorem elmnc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mnc 27314 . . . . 5 Poly coeffdeg
21dmmptss 5366 . . . 4
3 elfvdm 5757 . . . 4
42, 3sseldi 3346 . . 3
54elpwid 3808 . 2
6 plybss 20113 . . 3 Poly
76adantr 452 . 2 Poly coeffdeg
8 cnex 9071 . . . . . 6
98elpw2 4364 . . . . 5
10 fveq2 5728 . . . . . . 7 Poly Poly
11 rabeq 2950 . . . . . . 7 Poly Poly Poly coeffdeg Poly coeffdeg
1210, 11syl 16 . . . . . 6 Poly coeffdeg Poly coeffdeg
13 fvex 5742 . . . . . . 7 Poly
1413rabex 4354 . . . . . 6 Poly coeffdeg
1512, 1, 14fvmpt 5806 . . . . 5 Poly coeffdeg
169, 15sylbir 205 . . . 4 Poly coeffdeg
1716eleq2d 2503 . . 3 Poly coeffdeg
18 fveq2 5728 . . . . . 6 coeff coeff
19 fveq2 5728 . . . . . 6 deg deg
2018, 19fveq12d 5734 . . . . 5 coeffdeg coeffdeg
2120eqeq1d 2444 . . . 4 coeffdeg coeffdeg
2221elrab 3092 . . 3 Poly coeffdeg Poly coeffdeg
2317, 22syl6bb 253 . 2 Poly coeffdeg
245, 7, 23pm5.21nii 343 1 Poly coeffdeg
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  crab 2709   wss 3320  cpw 3799   cdm 4878  cfv 5454  cc 8988  c1 8991  Polycply 20103  coeffccoe 20105  degcdgr 20106   cmnc 27312 This theorem is referenced by:  mncply  27319  mnccoe  27320 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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-cnex 9046 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  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-fv 5462  df-ply 20107  df-mnc 27314
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