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Theorem ply1val 16547
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 5687 . . . . . 6  |-  ( r  =  R  ->  (PwSer1 `  r )  =  (PwSer1 `  R ) )
3 ply1val.2 . . . . . 6  |-  S  =  (PwSer1 `  R )
42, 3syl6eqr 2454 . . . . 5  |-  ( r  =  R  ->  (PwSer1 `  r )  =  S )
5 oveq2 6048 . . . . . 6  |-  ( r  =  R  ->  ( 1o mPoly  r )  =  ( 1o mPoly  R ) )
65fveq2d 5691 . . . . 5  |-  ( r  =  R  ->  ( Base `  ( 1o mPoly  r
) )  =  (
Base `  ( 1o mPoly  R ) ) )
74, 6oveq12d 6058 . . . 4  |-  ( r  =  R  ->  (
(PwSer1 `
 r )s  ( Base `  ( 1o mPoly  r )
) )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
8 df-ply1 16533 . . . 4  |- Poly1  =  (
r  e.  _V  |->  ( (PwSer1 `  r )s  ( Base `  ( 1o mPoly  r )
) ) )
9 ovex 6065 . . . 4  |-  ( Ss  (
Base `  ( 1o mPoly  R ) ) )  e. 
_V
107, 8, 9fvmpt 5765 . . 3  |-  ( R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
11 fvprc 5681 . . . . 5  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
12 ress0 13478 . . . . 5  |-  ( (/)s  ( Base `  ( 1o mPoly  R )
) )  =  (/)
1311, 12syl6eqr 2454 . . . 4  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (
(/)s 
( Base `  ( 1o mPoly  R ) ) ) )
14 fvprc 5681 . . . . . 6  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
153, 14syl5eq 2448 . . . . 5  |-  ( -.  R  e.  _V  ->  S  =  (/) )
1615oveq1d 6055 . . . 4  |-  ( -.  R  e.  _V  ->  ( Ss  ( Base `  ( 1o mPoly  R ) ) )  =  ( (/)s  ( Base `  ( 1o mPoly  R ) ) ) )
1713, 16eqtr4d 2439 . . 3  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
1810, 17pm2.61i 158 . 2  |-  (Poly1 `  R
)  =  ( Ss  (
Base `  ( 1o mPoly  R ) ) )
191, 18eqtri 2424 1  |-  P  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   ` cfv 5413  (class class class)co 6040   1oc1o 6676   Basecbs 13424   ↾s cress 13425   mPoly cmpl 16363  PwSer1cps1 16524  Poly1cpl1 16526
This theorem is referenced by:  ply1bas  16548  ply1crng  16551  ply1assa  16552  ply1bascl  16556  ply1plusg  16574  ply1vsca  16575  ply1mulr  16576  ply1rng  16597  ply1lmod  16601  ply1sca  16602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-slot 13428  df-base 13429  df-ress 13431  df-ply1 16533
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