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Theorem isuc1p 21624
Description: Being a unitic 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 )
uc1pval.c  |-  C  =  (Unic1p `  R )
uc1pval.u  |-  U  =  (Unit `  R )
Assertion
Ref Expression
isuc1p  |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/= 
.0.  /\  ( (coe1 `  F ) `  ( D `  F )
)  e.  U ) )

Proof of Theorem isuc1p
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 neeq1 2628 . . . 4  |-  ( f  =  F  ->  (
f  =/=  .0.  <->  F  =/=  .0.  ) )
2 fveq2 5703 . . . . . 6  |-  ( f  =  F  ->  (coe1 `  f )  =  (coe1 `  F ) )
3 fveq2 5703 . . . . . 6  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
42, 3fveq12d 5709 . . . . 5  |-  ( f  =  F  ->  (
(coe1 `  f ) `  ( D `  f ) )  =  ( (coe1 `  F ) `  ( D `  F )
) )
54eleq1d 2509 . . . 4  |-  ( f  =  F  ->  (
( (coe1 `  f ) `  ( D `  f ) )  e.  U  <->  ( (coe1 `  F ) `  ( D `  F )
)  e.  U ) )
61, 5anbi12d 710 . . 3  |-  ( f  =  F  ->  (
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U )  <-> 
( F  =/=  .0.  /\  ( (coe1 `  F ) `  ( D `  F ) )  e.  U ) ) )
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 uc1pval.c . . . 4  |-  C  =  (Unic1p `  R )
12 uc1pval.u . . . 4  |-  U  =  (Unit `  R )
137, 8, 9, 10, 11, 12uc1pval 21623 . . 3  |-  C  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) }
146, 13elrab2 3131 . 2  |-  ( F  e.  C  <->  ( F  e.  B  /\  ( F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  e.  U ) ) )
15 3anass 969 . 2  |-  ( ( F  e.  B  /\  F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  e.  U )  <-> 
( F  e.  B  /\  ( F  =/=  .0.  /\  ( (coe1 `  F ) `  ( D `  F ) )  e.  U ) ) )
1614, 15bitr4i 252 1  |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/= 
.0.  /\  ( (coe1 `  F ) `  ( D `  F )
)  e.  U ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   ` cfv 5430   Basecbs 14186   0gc0g 14390  Unitcui 16743  Poly1cpl1 17645  coe1cco1 17646   deg1 cdg1 21535  Unic1pcuc1p 21610
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-slot 14190  df-base 14191  df-uc1p 21615
This theorem is referenced by:  uc1pcl  21627  uc1pn0  21629  uc1pldg  21632  mon1puc1p  21634  drnguc1p  21654
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