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Theorem ply1val 16372
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 5608 . . . . . 6  |-  ( r  =  R  ->  (PwSer1 `  r )  =  (PwSer1 `  R ) )
3 ply1val.2 . . . . . 6  |-  S  =  (PwSer1 `  R )
42, 3syl6eqr 2408 . . . . 5  |-  ( r  =  R  ->  (PwSer1 `  r )  =  S )
5 oveq2 5953 . . . . . 6  |-  ( r  =  R  ->  ( 1o mPoly  r )  =  ( 1o mPoly  R ) )
65fveq2d 5612 . . . . 5  |-  ( r  =  R  ->  ( Base `  ( 1o mPoly  r
) )  =  (
Base `  ( 1o mPoly  R ) ) )
74, 6oveq12d 5963 . . . 4  |-  ( r  =  R  ->  (
(PwSer1 `
 r )s  ( Base `  ( 1o mPoly  r )
) )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
8 df-ply1 16358 . . . 4  |- Poly1  =  (
r  e.  _V  |->  ( (PwSer1 `  r )s  ( Base `  ( 1o mPoly  r )
) ) )
9 ovex 5970 . . . 4  |-  ( Ss  (
Base `  ( 1o mPoly  R ) ) )  e. 
_V
107, 8, 9fvmpt 5685 . . 3  |-  ( R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
11 fvprc 5602 . . . . 5  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
12 ress0 13299 . . . . 5  |-  ( (/)s  ( Base `  ( 1o mPoly  R )
) )  =  (/)
1311, 12syl6eqr 2408 . . . 4  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (
(/)s 
( Base `  ( 1o mPoly  R ) ) ) )
14 fvprc 5602 . . . . . 6  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
153, 14syl5eq 2402 . . . . 5  |-  ( -.  R  e.  _V  ->  S  =  (/) )
1615oveq1d 5960 . . . 4  |-  ( -.  R  e.  _V  ->  ( Ss  ( Base `  ( 1o mPoly  R ) ) )  =  ( (/)s  ( Base `  ( 1o mPoly  R ) ) ) )
1713, 16eqtr4d 2393 . . 3  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) ) )
1810, 17pm2.61i 156 . 2  |-  (Poly1 `  R
)  =  ( Ss  (
Base `  ( 1o mPoly  R ) ) )
191, 18eqtri 2378 1  |-  P  =  ( Ss  ( Base `  ( 1o mPoly  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   ` cfv 5337  (class class class)co 5945   1oc1o 6559   Basecbs 13245   ↾s cress 13246   mPoly cmpl 16188  PwSer1cps1 16349  Poly1cpl1 16351
This theorem is referenced by:  ply1bas  16373  ply1crng  16376  ply1assa  16377  ply1bascl  16383  ply1plusg  16402  ply1vsca  16403  ply1mulr  16404  ply1rng  16425  ply1lmod  16429  ply1sca  16430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-slot 13249  df-base 13250  df-ress 13252  df-ply1 16358
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