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Theorem ply1val 16597
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 5731 . . . . . 6  |-  ( r  =  R  ->  (PwSer1 `  r )  =  (PwSer1 `  R ) )
3 ply1val.2 . . . . . 6  |-  S  =  (PwSer1 `  R )
42, 3syl6eqr 2488 . . . . 5  |-  ( r  =  R  ->  (PwSer1 `  r )  =  S )
5 oveq2 6092 . . . . . 6  |-  ( r  =  R  ->  ( 1o mPoly  r )  =  ( 1o mPoly  R ) )
65fveq2d 5735 . . . . 5  |-  ( r  =  R  ->  ( Base `  ( 1o mPoly  r
) )  =  (
Base `  ( 1o mPoly  R ) ) )
74, 6oveq12d 6102 . . . 4  |-  ( r  =  R  ->  (
(PwSer1 `
 r )s  ( Base `  ( 1o mPoly  r )
) )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
8 df-ply1 16583 . . . 4  |- Poly1  =  (
r  e.  _V  |->  ( (PwSer1 `  r )s  ( Base `  ( 1o mPoly  r )
) ) )
9 ovex 6109 . . . 4  |-  ( Ss  (
Base `  ( 1o mPoly  R ) ) )  e. 
_V
107, 8, 9fvmpt 5809 . . 3  |-  ( R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
11 fvprc 5725 . . . . 5  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
12 ress0 13528 . . . . 5  |-  ( (/)s  ( Base `  ( 1o mPoly  R )
) )  =  (/)
1311, 12syl6eqr 2488 . . . 4  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (
(/)s 
( Base `  ( 1o mPoly  R ) ) ) )
14 fvprc 5725 . . . . . 6  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
153, 14syl5eq 2482 . . . . 5  |-  ( -.  R  e.  _V  ->  S  =  (/) )
1615oveq1d 6099 . . . 4  |-  ( -.  R  e.  _V  ->  ( Ss  ( Base `  ( 1o mPoly  R ) ) )  =  ( (/)s  ( Base `  ( 1o mPoly  R ) ) ) )
1713, 16eqtr4d 2473 . . 3  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
1810, 17pm2.61i 159 . 2  |-  (Poly1 `  R
)  =  ( Ss  (
Base `  ( 1o mPoly  R ) ) )
191, 18eqtri 2458 1  |-  P  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   ` cfv 5457  (class class class)co 6084   1oc1o 6720   Basecbs 13474   ↾s cress 13475   mPoly cmpl 16413  PwSer1cps1 16574  Poly1cpl1 16576
This theorem is referenced by:  ply1bas  16598  ply1crng  16601  ply1assa  16602  ply1bascl  16606  ply1plusg  16624  ply1vsca  16625  ply1mulr  16626  ply1rng  16647  ply1lmod  16651  ply1sca  16652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-slot 13478  df-base 13479  df-ress 13481  df-ply1 16583
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