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Theorem ismon1p 19528
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 2454 . . . 4  |-  ( f  =  F  ->  (
f  =/=  .0.  <->  F  =/=  .0.  ) )
2 fveq2 5525 . . . . . 6  |-  ( f  =  F  ->  (coe1 `  f )  =  (coe1 `  F ) )
3 fveq2 5525 . . . . . 6  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
42, 3fveq12d 5531 . . . . 5  |-  ( f  =  F  ->  (
(coe1 `  f ) `  ( D `  f ) )  =  ( (coe1 `  F ) `  ( D `  F )
) )
54eqeq1d 2291 . . . 4  |-  ( f  =  F  ->  (
( (coe1 `  f ) `  ( D `  f ) )  =  .1.  <->  ( (coe1 `  F ) `  ( D `  F )
)  =  .1.  )
)
61, 5anbi12d 691 . . 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 19527 . . 3  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
146, 13elrab2 2925 . 2  |-  ( F  e.  M  <->  ( F  e.  B  /\  ( F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  =  .1.  )
) )
15 3anass 938 . 2  |-  ( ( F  e.  B  /\  F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  =  .1.  )  <->  ( F  e.  B  /\  ( F  =/=  .0.  /\  ( (coe1 `  F ) `  ( D `  F ) )  =  .1.  )
) )
1614, 15bitr4i 243 1  |-  ( F  e.  M  <->  ( F  e.  B  /\  F  =/= 
.0.  /\  ( (coe1 `  F ) `  ( D `  F )
)  =  .1.  )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255   Basecbs 13148   0gc0g 13400   1rcur 15339  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440  Monic1pcmn1 19511
This theorem is referenced by:  mon1pcl  19530  mon1pn0  19532  mon1pldg  19535  uc1pmon1p  19537  ply1remlem  19548  mon1pid  27524  mon1psubm  27525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152  df-base 13153  df-mon1 19516
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