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Theorem isuc1p 22407
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 2748 . . . 4  |-  ( f  =  F  ->  (
f  =/=  .0.  <->  F  =/=  .0.  ) )
2 fveq2 5872 . . . . . 6  |-  ( f  =  F  ->  (coe1 `  f )  =  (coe1 `  F ) )
3 fveq2 5872 . . . . . 6  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
42, 3fveq12d 5878 . . . . 5  |-  ( f  =  F  ->  (
(coe1 `  f ) `  ( D `  f ) )  =  ( (coe1 `  F ) `  ( D `  F )
) )
54eleq1d 2536 . . . 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 22406 . . 3  |-  C  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) }
146, 13elrab2 3268 . 2  |-  ( F  e.  C  <->  ( F  e.  B  /\  ( F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  e.  U ) ) )
15 3anass 977 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5594   Basecbs 14506   0gc0g 14711  Unitcui 17158  Poly1cpl1 18084  coe1cco1 18085   deg1 cdg1 22318  Unic1pcuc1p 22393
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-slot 14510  df-base 14511  df-uc1p 22398
This theorem is referenced by:  uc1pcl  22410  uc1pn0  22412  uc1pldg  22415  mon1puc1p  22417  drnguc1p  22437
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