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Theorem elmnc 29498
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 29494 . . . . 5  |-  Monic  =  ( s  e.  ~P CC  |->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
21dmmptss 5339 . . . 4  |-  dom  Monic  C_ 
~P CC
3 elfvdm 5721 . . . 4  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  dom  Monic  )
42, 3sseldi 3359 . . 3  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  ~P CC )
54elpwid 3875 . 2  |-  ( P  e.  (  Monic  `  S
)  ->  S  C_  CC )
6 plybss 21667 . . 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 9368 . . . . . 6  |-  CC  e.  _V
98elpw2 4461 . . . . 5  |-  ( S  e.  ~P CC  <->  S  C_  CC )
10 fveq2 5696 . . . . . . 7  |-  ( s  =  S  ->  (Poly `  s )  =  (Poly `  S ) )
11 rabeq 2971 . . . . . . 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 5706 . . . . . . 7  |-  (Poly `  S )  e.  _V
1413rabex 4448 . . . . . 6  |-  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  e.  _V
1512, 1, 14fvmpt 5779 . . . . 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 2510 . . 3  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 } ) )
18 fveq2 5696 . . . . . 6  |-  ( p  =  P  ->  (coeff `  p )  =  (coeff `  P ) )
19 fveq2 5696 . . . . . 6  |-  ( p  =  P  ->  (deg `  p )  =  (deg
`  P ) )
2018, 19fveq12d 5702 . . . . 5  |-  ( p  =  P  ->  (
(coeff `  p ) `  (deg `  p )
)  =  ( (coeff `  P ) `  (deg `  P ) ) )
2120eqeq1d 2451 . . . 4  |-  ( p  =  P  ->  (
( (coeff `  p
) `  (deg `  p
) )  =  1  <-> 
( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2221elrab 3122 . . 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 1369    e. wcel 1756   {crab 2724    C_ wss 3333   ~Pcpw 3865   dom cdm 4845   ` cfv 5423   CCcc 9285   1c1 9288  Polycply 21657  coeffccoe 21659  degcdgr 21660    Monic cmnc 29492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-cnex 9343
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fv 5431  df-ply 21661  df-mnc 29494
This theorem is referenced by:  mncply  29499  mnccoe  29500
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