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Theorem mncply 40575
Description: A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Assertion
Ref Expression
mncply (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))

Proof of Theorem mncply
StepHypRef Expression
1 elmnc 40574 . 2 (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
21simplbi 501 1 (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cfv 6350  1c1 10629  Polycply 24946  coeffccoe 24948  degcdgr 24949   Monic cmnc 40569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-cnex 10684
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fv 6358  df-ply 24950  df-mnc 40571
This theorem is referenced by: (None)
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