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Theorem elmnc 30718
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 30714 . . . . 5  |-  Monic  =  ( s  e.  ~P CC  |->  { p  e.  (Poly `  s )  |  ( (coeff `  p ) `  (deg `  p )
)  =  1 } )
21dmmptss 5503 . . . 4  |-  dom  Monic  C_ 
~P CC
3 elfvdm 5892 . . . 4  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  dom  Monic  )
42, 3sseldi 3502 . . 3  |-  ( P  e.  (  Monic  `  S
)  ->  S  e.  ~P CC )
54elpwid 4020 . 2  |-  ( P  e.  (  Monic  `  S
)  ->  S  C_  CC )
6 plybss 22354 . . 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 9573 . . . . . 6  |-  CC  e.  _V
98elpw2 4611 . . . . 5  |-  ( S  e.  ~P CC  <->  S  C_  CC )
10 fveq2 5866 . . . . . . 7  |-  ( s  =  S  ->  (Poly `  s )  =  (Poly `  S ) )
11 rabeq 3107 . . . . . . 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 5876 . . . . . . 7  |-  (Poly `  S )  e.  _V
1413rabex 4598 . . . . . 6  |-  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 }  e.  _V
1512, 1, 14fvmpt 5950 . . . . 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 2537 . . 3  |-  ( S 
C_  CC  ->  ( P  e.  (  Monic  `  S
)  <->  P  e.  { p  e.  (Poly `  S )  |  ( (coeff `  p ) `  (deg `  p ) )  =  1 } ) )
18 fveq2 5866 . . . . . 6  |-  ( p  =  P  ->  (coeff `  p )  =  (coeff `  P ) )
19 fveq2 5866 . . . . . 6  |-  ( p  =  P  ->  (deg `  p )  =  (deg
`  P ) )
2018, 19fveq12d 5872 . . . . 5  |-  ( p  =  P  ->  (
(coeff `  p ) `  (deg `  p )
)  =  ( (coeff `  P ) `  (deg `  P ) ) )
2120eqeq1d 2469 . . . 4  |-  ( p  =  P  ->  (
( (coeff `  p
) `  (deg `  p
) )  =  1  <-> 
( (coeff `  P
) `  (deg `  P
) )  =  1 ) )
2221elrab 3261 . . 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 1379    e. wcel 1767   {crab 2818    C_ wss 3476   ~Pcpw 4010   dom cdm 4999   ` cfv 5588   CCcc 9490   1c1 9493  Polycply 22344  coeffccoe 22346  degcdgr 22347    Monic cmnc 30712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-cnex 9548
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ply 22348  df-mnc 30714
This theorem is referenced by:  mncply  30719  mnccoe  30720
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