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Theorem isuc1p 19542
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 2467 . . . 4  |-  ( f  =  F  ->  (
f  =/=  .0.  <->  F  =/=  .0.  ) )
2 fveq2 5541 . . . . . 6  |-  ( f  =  F  ->  (coe1 `  f )  =  (coe1 `  F ) )
3 fveq2 5541 . . . . . 6  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
42, 3fveq12d 5547 . . . . 5  |-  ( f  =  F  ->  (
(coe1 `  f ) `  ( D `  f ) )  =  ( (coe1 `  F ) `  ( D `  F )
) )
54eleq1d 2362 . . . 4  |-  ( f  =  F  ->  (
( (coe1 `  f ) `  ( D `  f ) )  e.  U  <->  ( (coe1 `  F ) `  ( D `  F )
)  e.  U ) )
61, 5anbi12d 691 . . 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 19541 . . 3  |-  C  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) }
146, 13elrab2 2938 . 2  |-  ( F  e.  C  <->  ( F  e.  B  /\  ( F  =/=  .0.  /\  (
(coe1 `  F ) `  ( D `  F ) )  e.  U ) ) )
15 3anass 938 . 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 243 1  |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/= 
.0.  /\  ( (coe1 `  F ) `  ( D `  F )
)  e.  U ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271   Basecbs 13164   0gc0g 13416  Unitcui 15437  Poly1cpl1 16268  coe1cco1 16271   deg1 cdg1 19456  Unic1pcuc1p 19528
This theorem is referenced by:  uc1pcl  19545  uc1pn0  19547  uc1pldg  19550  mon1puc1p  19552  drnguc1p  19572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168  df-base 13169  df-uc1p 19533
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