MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uc1pval Structured version   Unicode version

Theorem uc1pval 22408
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 5872 . . . . . . . 8  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
3 uc1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
42, 3syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
54fveq2d 5876 . . . . . 6  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
6 uc1pval.b . . . . . 6  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2526 . . . . 5  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
84fveq2d 5876 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
9 uc1pval.z . . . . . . . 8  |-  .0.  =  ( 0g `  P )
108, 9syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1110neeq2d 2745 . . . . . 6  |-  ( r  =  R  ->  (
f  =/=  ( 0g
`  (Poly1 `  r ) )  <-> 
f  =/=  .0.  )
)
12 fveq2 5872 . . . . . . . . . 10  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
13 uc1pval.d . . . . . . . . . 10  |-  D  =  ( deg1  `  R )
1412, 13syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1514fveq1d 5874 . . . . . . . 8  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  f )  =  ( D `  f ) )
1615fveq2d 5876 . . . . . . 7  |-  ( r  =  R  ->  (
(coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( (coe1 `  f ) `  ( D `  f ) ) )
17 fveq2 5872 . . . . . . . 8  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
18 uc1pval.u . . . . . . . 8  |-  U  =  (Unit `  R )
1917, 18syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  (Unit `  r )  =  U )
2016, 19eleq12d 2549 . . . . . 6  |-  ( r  =  R  ->  (
( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  e.  (Unit `  r )  <->  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) )
2111, 20anbi12d 710 . . . . 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 3113 . . . 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 22400 . . . 4  |- Unic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
24 fvex 5882 . . . . . 6  |-  ( Base `  P )  e.  _V
256, 24eqeltri 2551 . . . . 5  |-  B  e. 
_V
2625rabex 4604 . . . 4  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  e.  _V
2722, 23, 26fvmpt 5957 . . 3  |-  ( R  e.  _V  ->  (Unic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) } )
28 fvprc 5866 . . . 4  |-  ( -.  R  e.  _V  ->  (Unic1p `  R )  =  (/) )
29 ssrab2 3590 . . . . . 6  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  C_  B
30 fvprc 5866 . . . . . . . . . 10  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
313, 30syl5eq 2520 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  P  =  (/) )
3231fveq2d 5876 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  ( Base `  (/) ) )
33 base0 14546 . . . . . . . 8  |-  (/)  =  (
Base `  (/) )
3432, 33syl6eqr 2526 . . . . . . 7  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  (/) )
356, 34syl5eq 2520 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3629, 35syl5sseq 3557 . . . . 5  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }  C_  (/) )
37 ss0 3821 . . . . 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 2511 . . 3  |-  ( -.  R  e.  _V  ->  (Unic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) } )
4027, 39pm2.61i 164 . 2  |-  (Unic1p `  R
)  =  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  e.  U ) }
411, 40eqtri 2496 1  |-  C  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  e.  U ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821   _Vcvv 3118    C_ wss 3481   (/)c0 3790   ` cfv 5594   Basecbs 14507   0gc0g 14712  Unitcui 17160  Poly1cpl1 18086  coe1cco1 18087   deg1 cdg1 22320  Unic1pcuc1p 22395
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 14511  df-base 14512  df-uc1p 22400
This theorem is referenced by:  isuc1p  22409
  Copyright terms: Public domain W3C validator