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Theorem ismon1p 21619
Description: Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pval.p  |-  P  =  (Poly1 `  R )
uc1pval.b  |-  B  =  ( Base `  P
)
uc1pval.z  |-  .0.  =  ( 0g `  P )
uc1pval.d  |-  D  =  ( deg1  `  R )
mon1pval.m  |-  M  =  (Monic1p `  R )
mon1pval.o  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
ismon1p  |-  ( F  e.  M  <->  ( F  e.  B  /\  F  =/= 
.0.  /\  ( (coe1 `  F ) `  ( D `  F )
)  =  .1.  )
)

Proof of Theorem ismon1p
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 neeq1 2621 . . . 4  |-  ( f  =  F  ->  (
f  =/=  .0.  <->  F  =/=  .0.  ) )
2 fveq2 5696 . . . . . 6  |-  ( f  =  F  ->  (coe1 `  f )  =  (coe1 `  F ) )
3 fveq2 5696 . . . . . 6  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
42, 3fveq12d 5702 . . . . 5  |-  ( f  =  F  ->  (
(coe1 `  f ) `  ( D `  f ) )  =  ( (coe1 `  F ) `  ( D `  F )
) )
54eqeq1d 2451 . . . 4  |-  ( f  =  F  ->  (
( (coe1 `  f ) `  ( D `  f ) )  =  .1.  <->  ( (coe1 `  F ) `  ( D `  F )
)  =  .1.  )
)
61, 5anbi12d 710 . . 3  |-  ( f  =  F  ->  (
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  )  <->  ( F  =/=  .0.  /\  ( (coe1 `  F ) `  ( D `  F ) )  =  .1.  )
) )
7 uc1pval.p . . . 4  |-  P  =  (Poly1 `  R )
8 uc1pval.b . . . 4  |-  B  =  ( Base `  P
)
9 uc1pval.z . . . 4  |-  .0.  =  ( 0g `  P )
10 uc1pval.d . . . 4  |-  D  =  ( deg1  `  R )
11 mon1pval.m . . . 4  |-  M  =  (Monic1p `  R )
12 mon1pval.o . . . 4  |-  .1.  =  ( 1r `  R )
137, 8, 9, 10, 11, 12mon1pval 21618 . . 3  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
146, 13elrab2 3124 . 2  |-  ( F  e.  M  <->  ( F  e.  B  /\  ( F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  =  .1.  )
) )
15 3anass 969 . 2  |-  ( ( F  e.  B  /\  F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  =  .1.  )  <->  ( F  e.  B  /\  ( F  =/=  .0.  /\  ( (coe1 `  F ) `  ( D `  F ) )  =  .1.  )
) )
1614, 15bitr4i 252 1  |-  ( F  e.  M  <->  ( F  e.  B  /\  F  =/= 
.0.  /\  ( (coe1 `  F ) `  ( D `  F )
)  =  .1.  )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   ` cfv 5423   Basecbs 14179   0gc0g 14383   1rcur 16608  Poly1cpl1 17638  coe1cco1 17639   deg1 cdg1 21528  Monic1pcmn1 21602
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
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-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-iota 5386  df-fun 5425  df-fv 5431  df-slot 14183  df-base 14184  df-mon1 21607
This theorem is referenced by:  mon1pcl  21621  mon1pn0  21623  mon1pldg  21626  uc1pmon1p  21628  ply1remlem  21639  mon1pid  29578  mon1psubm  29579
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