MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ply1val Structured version   Unicode version

Theorem ply1val 18032
Description: The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypotheses
Ref Expression
ply1val.1  |-  P  =  (Poly1 `  R )
ply1val.2  |-  S  =  (PwSer1 `  R )
Assertion
Ref Expression
ply1val  |-  P  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) )

Proof of Theorem ply1val
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 ply1val.1 . 2  |-  P  =  (Poly1 `  R )
2 fveq2 5866 . . . . . 6  |-  ( r  =  R  ->  (PwSer1 `  r )  =  (PwSer1 `  R ) )
3 ply1val.2 . . . . . 6  |-  S  =  (PwSer1 `  R )
42, 3syl6eqr 2526 . . . . 5  |-  ( r  =  R  ->  (PwSer1 `  r )  =  S )
5 oveq2 6292 . . . . . 6  |-  ( r  =  R  ->  ( 1o mPoly  r )  =  ( 1o mPoly  R ) )
65fveq2d 5870 . . . . 5  |-  ( r  =  R  ->  ( Base `  ( 1o mPoly  r
) )  =  (
Base `  ( 1o mPoly  R ) ) )
74, 6oveq12d 6302 . . . 4  |-  ( r  =  R  ->  (
(PwSer1 `
 r )s  ( Base `  ( 1o mPoly  r )
) )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
8 df-ply1 18020 . . . 4  |- Poly1  =  (
r  e.  _V  |->  ( (PwSer1 `  r )s  ( Base `  ( 1o mPoly  r )
) ) )
9 ovex 6309 . . . 4  |-  ( Ss  (
Base `  ( 1o mPoly  R ) ) )  e. 
_V
107, 8, 9fvmpt 5950 . . 3  |-  ( R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
11 fvprc 5860 . . . . 5  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
12 ress0 14549 . . . . 5  |-  ( (/)s  ( Base `  ( 1o mPoly  R )
) )  =  (/)
1311, 12syl6eqr 2526 . . . 4  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (
(/)s 
( Base `  ( 1o mPoly  R ) ) ) )
14 fvprc 5860 . . . . . 6  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
153, 14syl5eq 2520 . . . . 5  |-  ( -.  R  e.  _V  ->  S  =  (/) )
1615oveq1d 6299 . . . 4  |-  ( -.  R  e.  _V  ->  ( Ss  ( Base `  ( 1o mPoly  R ) ) )  =  ( (/)s  ( Base `  ( 1o mPoly  R ) ) ) )
1713, 16eqtr4d 2511 . . 3  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
1810, 17pm2.61i 164 . 2  |-  (Poly1 `  R
)  =  ( Ss  (
Base `  ( 1o mPoly  R ) ) )
191, 18eqtri 2496 1  |-  P  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ` cfv 5588  (class class class)co 6284   1oc1o 7123   Basecbs 14490   ↾s cress 14491   mPoly cmpl 17801  PwSer1cps1 18013  Poly1cpl1 18015
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-slot 14494  df-base 14495  df-ress 14497  df-ply1 18020
This theorem is referenced by:  ply1bas  18033  ply1crng  18036  ply1assa  18037  ply1bascl  18041  ply1plusg  18065  ply1vsca  18066  ply1mulr  18067  ply1rng  18088  ply1lmod  18092  ply1sca  18093
  Copyright terms: Public domain W3C validator