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Theorem uc1pval 23169
Description: Value of the set of unitic polynomials. (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
uc1pval  |-  C  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) }
Distinct variable groups:    B, f    D, f    R, f    U, f    .0. , f
Allowed substitution hints:    C( f)    P( f)

Proof of Theorem uc1pval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 uc1pval.c . 2  |-  C  =  (Unic1p `  R )
2 fveq2 5879 . . . . . . . 8  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
3 uc1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
42, 3syl6eqr 2523 . . . . . . 7  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
54fveq2d 5883 . . . . . 6  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
6 uc1pval.b . . . . . 6  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2523 . . . . 5  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
84fveq2d 5883 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
9 uc1pval.z . . . . . . . 8  |-  .0.  =  ( 0g `  P )
108, 9syl6eqr 2523 . . . . . . 7  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1110neeq2d 2703 . . . . . 6  |-  ( r  =  R  ->  (
f  =/=  ( 0g
`  (Poly1 `  r ) )  <-> 
f  =/=  .0.  )
)
12 fveq2 5879 . . . . . . . . . 10  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
13 uc1pval.d . . . . . . . . . 10  |-  D  =  ( deg1  `  R )
1412, 13syl6eqr 2523 . . . . . . . . 9  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1514fveq1d 5881 . . . . . . . 8  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  f )  =  ( D `  f ) )
1615fveq2d 5883 . . . . . . 7  |-  ( r  =  R  ->  (
(coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( (coe1 `  f ) `  ( D `  f ) ) )
17 fveq2 5879 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
18 uc1pval.u . . . . . . . 8  |-  U  =  (Unit `  R )
1917, 18syl6eqr 2523 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
2016, 19eleq12d 2543 . . . . . 6  |-  ( r  =  R  ->  (
( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  e.  (Unit `  r )  <->  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) )
2111, 20anbi12d 725 . . . . 5  |-  ( r  =  R  ->  (
( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) )  <->  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) ) )
227, 21rabeqbidv 3026 . . . 4  |-  ( r  =  R  ->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) }  =  { f  e.  B  |  ( f  =/=  .0.  /\  (
(coe1 `  f ) `  ( D `  f ) )  e.  U ) } )
23 df-uc1p 23160 . . . 4  |- Unic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
24 fvex 5889 . . . . . 6  |-  ( Base `  P )  e.  _V
256, 24eqeltri 2545 . . . . 5  |-  B  e. 
_V
2625rabex 4550 . . . 4  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  e.  _V
2722, 23, 26fvmpt 5963 . . 3  |-  ( R  e.  _V  ->  (Unic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) } )
28 fvprc 5873 . . . 4  |-  ( -.  R  e.  _V  ->  (Unic1p `  R )  =  (/) )
29 ssrab2 3500 . . . . . 6  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  C_  B
30 fvprc 5873 . . . . . . . . . 10  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
313, 30syl5eq 2517 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  P  =  (/) )
3231fveq2d 5883 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  ( Base `  (/) ) )
33 base0 15240 . . . . . . . 8  |-  (/)  =  (
Base `  (/) )
3432, 33syl6eqr 2523 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  (/) )
356, 34syl5eq 2517 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3629, 35syl5sseq 3466 . . . . 5  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  C_  (/) )
37 ss0 3768 . . . . 5  |-  ( { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  C_  (/)  ->  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  =  (/) )
3836, 37syl 17 . . . 4  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  =  (/) )
3928, 38eqtr4d 2508 . . 3  |-  ( -.  R  e.  _V  ->  (Unic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) } )
4027, 39pm2.61i 169 . 2  |-  (Unic1p `  R
)  =  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }
411, 40eqtri 2493 1  |-  C  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ` cfv 5589   Basecbs 15199   0gc0g 15416  Unitcui 17945  Poly1cpl1 18847  coe1cco1 18848   deg1 cdg1 23082  Unic1pcuc1p 23154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-slot 15203  df-base 15204  df-uc1p 23160
This theorem is referenced by:  isuc1p  23170
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