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Theorem elmnc 31289
 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 31286 . . . . 5 Poly coeffdeg
21dmmptss 5509 . . . 4
3 elfvdm 5898 . . . 4
42, 3sseldi 3497 . . 3
54elpwid 4025 . 2
6 plybss 22717 . . 3 Poly
76adantr 465 . 2 Poly coeffdeg
8 cnex 9590 . . . . . 6
98elpw2 4620 . . . . 5
10 fveq2 5872 . . . . . . 7 Poly Poly
11 rabeq 3103 . . . . . . 7 Poly Poly Poly coeffdeg Poly coeffdeg
1210, 11syl 16 . . . . . 6 Poly coeffdeg Poly coeffdeg
13 fvex 5882 . . . . . . 7 Poly
1413rabex 4607 . . . . . 6 Poly coeffdeg
1512, 1, 14fvmpt 5956 . . . . 5 Poly coeffdeg
169, 15sylbir 213 . . . 4 Poly coeffdeg
1716eleq2d 2527 . . 3 Poly coeffdeg
18 fveq2 5872 . . . . . 6 coeff coeff
19 fveq2 5872 . . . . . 6 deg deg
2018, 19fveq12d 5878 . . . . 5 coeffdeg coeffdeg
2120eqeq1d 2459 . . . 4 coeffdeg coeffdeg
2221elrab 3257 . . 3 Poly coeffdeg Poly coeffdeg
2317, 22syl6bb 261 . 2 Poly coeffdeg
245, 7, 23pm5.21nii 353 1 Poly coeffdeg
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1395   wcel 1819  crab 2811   wss 3471  cpw 4015   cdm 5008  cfv 5594  cc 9507  c1 9510  Polycply 22707  coeffccoe 22709  degcdgr 22710   cmnc 31284 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-cnex 9565 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ply 22711  df-mnc 31286 This theorem is referenced by:  mncply  31290  mnccoe  31291
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