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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  |-  ( 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 31286 . . . . 5  |-  Monic  =  ( s  e.  ~P CC  |->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
21dmmptss 5509 . . . 4  |-  dom  Monic  C_ 
~P CC
3 elfvdm 5898 . . . 4  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  dom  Monic  )
42, 3sseldi 3497 . . 3  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  ~P CC )
54elpwid 4025 . 2  |-  ( P  e.  (  Monic  `  S
)  ->  S  C_  CC )
6 plybss 22717 . . 3  |-  ( P  e.  (Poly `  S
)  ->  S  C_  CC )
76adantr 465 . 2  |-  ( ( P  e.  (Poly `  S )  /\  (
(coeff `  P ) `  (deg `  P )
)  =  1 )  ->  S  C_  CC )
8 cnex 9590 . . . . . 6  |-  CC  e.  _V
98elpw2 4620 . . . . 5  |-  ( S  e.  ~P CC  <->  S  C_  CC )
10 fveq2 5872 . . . . . . 7  |-  ( s  =  S  ->  (Poly `  s )  =  (Poly `  S ) )
11 rabeq 3103 . . . . . . 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 5882 . . . . . . 7  |-  (Poly `  S )  e.  _V
1413rabex 4607 . . . . . 6  |-  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  e.  _V
1512, 1, 14fvmpt 5956 . . . . 5  |-  ( S  e.  ~P CC  ->  ( 
Monic  `  S )  =  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
169, 15sylbir 213 . . . 4  |-  ( S 
C_  CC  ->  (  Monic  `  S )  =  {
p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
1716eleq2d 2527 . . 3  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 } ) )
18 fveq2 5872 . . . . . 6  |-  ( p  =  P  ->  (coeff `  p )  =  (coeff `  P ) )
19 fveq2 5872 . . . . . 6  |-  ( p  =  P  ->  (deg `  p )  =  (deg
`  P ) )
2018, 19fveq12d 5878 . . . . 5  |-  ( p  =  P  ->  (
(coeff `  p ) `  (deg `  p )
)  =  ( (coeff `  P ) `  (deg `  P ) ) )
2120eqeq1d 2459 . . . 4  |-  ( p  =  P  ->  (
( (coeff `  p
) `  (deg `  p
) )  =  1  <-> 
( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2221elrab 3257 . . 3  |-  ( P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2317, 22syl6bb 261 . 2  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) ) )
245, 7, 23pm5.21nii 353 1  |-  ( P  e.  (  Monic  `  S
)  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811    C_ wss 3471   ~Pcpw 4015   dom cdm 5008   ` cfv 5594   CCcc 9507   1c1 9510  Polycply 22707  coeffccoe 22709  degcdgr 22710    Monic 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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