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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  |-  ( 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 27314 . . . . 5  |-  Monic  =  ( s  e.  ~P CC  |->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
21dmmptss 5366 . . . 4  |-  dom  Monic  C_ 
~P CC
3 elfvdm 5757 . . . 4  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  dom  Monic  )
42, 3sseldi 3346 . . 3  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  ~P CC )
54elpwid 3808 . 2  |-  ( P  e.  (  Monic  `  S
)  ->  S  C_  CC )
6 plybss 20113 . . 3  |-  ( P  e.  (Poly `  S
)  ->  S  C_  CC )
76adantr 452 . 2  |-  ( ( P  e.  (Poly `  S )  /\  (
(coeff `  P ) `  (deg `  P )
)  =  1 )  ->  S  C_  CC )
8 cnex 9071 . . . . . 6  |-  CC  e.  _V
98elpw2 4364 . . . . 5  |-  ( S  e.  ~P CC  <->  S  C_  CC )
10 fveq2 5728 . . . . . . 7  |-  ( s  =  S  ->  (Poly `  s )  =  (Poly `  S ) )
11 rabeq 2950 . . . . . . 7  |-  ( (Poly `  s )  =  (Poly `  S )  ->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  =  {
p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
1210, 11syl 16 . . . . . 6  |-  ( s  =  S  ->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  =  {
p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
13 fvex 5742 . . . . . . 7  |-  (Poly `  S )  e.  _V
1413rabex 4354 . . . . . 6  |-  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  e.  _V
1512, 1, 14fvmpt 5806 . . . . 5  |-  ( S  e.  ~P CC  ->  ( 
Monic  `  S )  =  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
169, 15sylbir 205 . . . 4  |-  ( S 
C_  CC  ->  (  Monic  `  S )  =  {
p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
1716eleq2d 2503 . . 3  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 } ) )
18 fveq2 5728 . . . . . 6  |-  ( p  =  P  ->  (coeff `  p )  =  (coeff `  P ) )
19 fveq2 5728 . . . . . 6  |-  ( p  =  P  ->  (deg `  p )  =  (deg
`  P ) )
2018, 19fveq12d 5734 . . . . 5  |-  ( p  =  P  ->  (
(coeff `  p ) `  (deg `  p )
)  =  ( (coeff `  P ) `  (deg `  P ) ) )
2120eqeq1d 2444 . . . 4  |-  ( p  =  P  ->  (
( (coeff `  p
) `  (deg `  p
) )  =  1  <-> 
( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2221elrab 3092 . . 3  |-  ( P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2317, 22syl6bb 253 . 2  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) ) )
245, 7, 23pm5.21nii 343 1  |-  ( P  e.  (  Monic  `  S
)  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709    C_ wss 3320   ~Pcpw 3799   dom cdm 4878   ` cfv 5454   CCcc 8988   1c1 8991  Polycply 20103  coeffccoe 20105  degcdgr 20106    Monic 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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