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Theorem ply1val 18431
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 5848 . . . . . 6  |-  ( r  =  R  ->  (PwSer1 `  r )  =  (PwSer1 `  R ) )
3 ply1val.2 . . . . . 6  |-  S  =  (PwSer1 `  R )
42, 3syl6eqr 2513 . . . . 5  |-  ( r  =  R  ->  (PwSer1 `  r )  =  S )
5 oveq2 6278 . . . . . 6  |-  ( r  =  R  ->  ( 1o mPoly  r )  =  ( 1o mPoly  R ) )
65fveq2d 5852 . . . . 5  |-  ( r  =  R  ->  ( Base `  ( 1o mPoly  r
) )  =  (
Base `  ( 1o mPoly  R ) ) )
74, 6oveq12d 6288 . . . 4  |-  ( r  =  R  ->  (
(PwSer1 `
 r )s  ( Base `  ( 1o mPoly  r )
) )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
8 df-ply1 18419 . . . 4  |- Poly1  =  (
r  e.  _V  |->  ( (PwSer1 `  r )s  ( Base `  ( 1o mPoly  r )
) ) )
9 ovex 6298 . . . 4  |-  ( Ss  (
Base `  ( 1o mPoly  R ) ) )  e. 
_V
107, 8, 9fvmpt 5931 . . 3  |-  ( R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
11 fvprc 5842 . . . . 5  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
12 ress0 14780 . . . . 5  |-  ( (/)s  ( Base `  ( 1o mPoly  R )
) )  =  (/)
1311, 12syl6eqr 2513 . . . 4  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (
(/)s 
( Base `  ( 1o mPoly  R ) ) ) )
14 fvprc 5842 . . . . . 6  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
153, 14syl5eq 2507 . . . . 5  |-  ( -.  R  e.  _V  ->  S  =  (/) )
1615oveq1d 6285 . . . 4  |-  ( -.  R  e.  _V  ->  ( Ss  ( Base `  ( 1o mPoly  R ) ) )  =  ( (/)s  ( Base `  ( 1o mPoly  R ) ) ) )
1713, 16eqtr4d 2498 . . 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 2483 1  |-  P  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   ` cfv 5570  (class class class)co 6270   1oc1o 7115   Basecbs 14719   ↾s cress 14720   mPoly cmpl 18200  PwSer1cps1 18412  Poly1cpl1 18414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-slot 14723  df-base 14724  df-ress 14726  df-ply1 18419
This theorem is referenced by:  ply1bas  18432  ply1crng  18435  ply1assa  18436  ply1bascl  18440  ply1plusg  18464  ply1vsca  18465  ply1mulr  18466  ply1ring  18487  ply1lmod  18491  ply1sca  18492
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