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Theorem uc1pval 20062
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 5728 . . . . . . . 8  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
3 uc1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
42, 3syl6eqr 2486 . . . . . . 7  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
54fveq2d 5732 . . . . . 6  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
6 uc1pval.b . . . . . 6  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2486 . . . . 5  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
84fveq2d 5732 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
9 uc1pval.z . . . . . . . 8  |-  .0.  =  ( 0g `  P )
108, 9syl6eqr 2486 . . . . . . 7  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1110neeq2d 2615 . . . . . 6  |-  ( r  =  R  ->  (
f  =/=  ( 0g
`  (Poly1 `  r ) )  <-> 
f  =/=  .0.  )
)
12 fveq2 5728 . . . . . . . . . 10  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
13 uc1pval.d . . . . . . . . . 10  |-  D  =  ( deg1  `  R )
1412, 13syl6eqr 2486 . . . . . . . . 9  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1514fveq1d 5730 . . . . . . . 8  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  f )  =  ( D `  f ) )
1615fveq2d 5732 . . . . . . 7  |-  ( r  =  R  ->  (
(coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( (coe1 `  f ) `  ( D `  f ) ) )
17 fveq2 5728 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
18 uc1pval.u . . . . . . . 8  |-  U  =  (Unit `  R )
1917, 18syl6eqr 2486 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
2016, 19eleq12d 2504 . . . . . 6  |-  ( r  =  R  ->  (
( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  e.  (Unit `  r )  <->  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) )
2111, 20anbi12d 692 . . . . 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 2951 . . . 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 20054 . . . 4  |- Unic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
24 fvex 5742 . . . . . 6  |-  ( Base `  P )  e.  _V
256, 24eqeltri 2506 . . . . 5  |-  B  e. 
_V
2625rabex 4354 . . . 4  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  e.  _V
2722, 23, 26fvmpt 5806 . . 3  |-  ( R  e.  _V  ->  (Unic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) } )
28 fvprc 5722 . . . 4  |-  ( -.  R  e.  _V  ->  (Unic1p `  R )  =  (/) )
29 ssrab2 3428 . . . . . 6  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  C_  B
30 fvprc 5722 . . . . . . . . . 10  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
313, 30syl5eq 2480 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  P  =  (/) )
3231fveq2d 5732 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  ( Base `  (/) ) )
33 base0 13506 . . . . . . . 8  |-  (/)  =  (
Base `  (/) )
3432, 33syl6eqr 2486 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  (/) )
356, 34syl5eq 2480 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3629, 35syl5sseq 3396 . . . . 5  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  C_  (/) )
37 ss0 3658 . . . . 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 16 . . . 4  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  =  (/) )
3928, 38eqtr4d 2471 . . 3  |-  ( -.  R  e.  _V  ->  (Unic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) } )
4027, 39pm2.61i 158 . 2  |-  (Unic1p `  R
)  =  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }
411, 40eqtri 2456 1  |-  C  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ` cfv 5454   Basecbs 13469   0gc0g 13723  Unitcui 15744  Poly1cpl1 16571  coe1cco1 16574   deg1 cdg1 19977  Unic1pcuc1p 20049
This theorem is referenced by:  isuc1p  20063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-slot 13473  df-base 13474  df-uc1p 20054
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