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Theorem elmnc 27341
Description: Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Assertion
Ref Expression
elmnc  |-  ( P  e.  (  Monic  `  S
)  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) )

Proof of Theorem elmnc
Dummy variables  s  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mnc 27337 . . . . 5  |-  Monic  =  ( s  e.  ~P CC  |->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
21dmmptss 5169 . . . 4  |-  dom  Monic  C_ 
~P CC
3 elfvdm 5554 . . . 4  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  dom  Monic  )
42, 3sseldi 3178 . . 3  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  ~P CC )
5 elpwi 3633 . . 3  |-  ( S  e.  ~P CC  ->  S 
C_  CC )
64, 5syl 15 . 2  |-  ( P  e.  (  Monic  `  S
)  ->  S  C_  CC )
7 plybss 19576 . . 3  |-  ( P  e.  (Poly `  S
)  ->  S  C_  CC )
87adantr 451 . 2  |-  ( ( P  e.  (Poly `  S )  /\  (
(coeff `  P ) `  (deg `  P )
)  =  1 )  ->  S  C_  CC )
9 cnex 8818 . . . . . 6  |-  CC  e.  _V
109elpw2 4175 . . . . 5  |-  ( S  e.  ~P CC  <->  S  C_  CC )
11 fveq2 5525 . . . . . . 7  |-  ( s  =  S  ->  (Poly `  s )  =  (Poly `  S ) )
12 rabeq 2782 . . . . . . 7  |-  ( (Poly `  s )  =  (Poly `  S )  ->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  =  {
p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
1311, 12syl 15 . . . . . 6  |-  ( s  =  S  ->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  =  {
p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
14 fvex 5539 . . . . . . 7  |-  (Poly `  S )  e.  _V
1514rabex 4165 . . . . . 6  |-  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  e.  _V
1613, 1, 15fvmpt 5602 . . . . 5  |-  ( S  e.  ~P CC  ->  ( 
Monic  `  S )  =  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
1710, 16sylbir 204 . . . 4  |-  ( S 
C_  CC  ->  (  Monic  `  S )  =  {
p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
1817eleq2d 2350 . . 3  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 } ) )
19 fveq2 5525 . . . . . 6  |-  ( p  =  P  ->  (coeff `  p )  =  (coeff `  P ) )
20 fveq2 5525 . . . . . 6  |-  ( p  =  P  ->  (deg `  p )  =  (deg
`  P ) )
2119, 20fveq12d 5531 . . . . 5  |-  ( p  =  P  ->  (
(coeff `  p ) `  (deg `  p )
)  =  ( (coeff `  P ) `  (deg `  P ) ) )
2221eqeq1d 2291 . . . 4  |-  ( p  =  P  ->  (
( (coeff `  p
) `  (deg `  p
) )  =  1  <-> 
( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2322elrab 2923 . . 3  |-  ( P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2418, 23syl6bb 252 . 2  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) ) )
256, 8, 24pm5.21nii 342 1  |-  ( P  e.  (  Monic  `  S
)  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ~Pcpw 3625   dom cdm 4689   ` cfv 5255   CCcc 8735   1c1 8738  Polycply 19566  coeffccoe 19568  degcdgr 19569    Monic cmnc 27335
This theorem is referenced by:  mncply  27342  mnccoe  27343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-cnex 8793
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ply 19570  df-mnc 27337
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