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Theorem mnccoe 42439
Description: A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Assertion
Ref Expression
mnccoe (𝑃 ∈ ( Monic β€˜π‘†) β†’ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1)

Proof of Theorem mnccoe
StepHypRef Expression
1 elmnc 42437 . 2 (𝑃 ∈ ( Monic β€˜π‘†) ↔ (𝑃 ∈ (Polyβ€˜π‘†) ∧ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1))
21simprbi 496 1 (𝑃 ∈ ( Monic β€˜π‘†) β†’ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  1c1 11110  Polycply 26069  coeffccoe 26071  degcdgr 26072   Monic cmnc 42432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-cnex 11165
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-ply 26073  df-mnc 42434
This theorem is referenced by:  mncn0  42440
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