MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psr1val Unicode version

Theorem psr1val 16281
Description: Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypothesis
Ref Expression
psr1val.1  |-  S  =  (PwSer1 `  R )
Assertion
Ref Expression
psr1val  |-  S  =  ( ( 1o ordPwSer  R ) `
 (/) )

Proof of Theorem psr1val
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 psr1val.1 . 2  |-  S  =  (PwSer1 `  R )
2 oveq2 5882 . . . . 5  |-  ( r  =  R  ->  ( 1o ordPwSer  r )  =  ( 1o ordPwSer  R ) )
32fveq1d 5543 . . . 4  |-  ( r  =  R  ->  (
( 1o ordPwSer  r ) `  (/) )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
4 df-psr1 16273 . . . 4  |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `  (/) ) )
5 fvex 5555 . . . 4  |-  ( ( 1o ordPwSer  R ) `  (/) )  e. 
_V
63, 4, 5fvmpt 5618 . . 3  |-  ( R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
7 fv01 5575 . . . . 5  |-  ( (/) `  (/) )  =  (/)
87eqcomi 2300 . . . 4  |-  (/)  =  (
(/) `  (/) )
9 fvprc 5535 . . . 4  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
10 reldmopsr 16231 . . . . . 6  |-  Rel  dom ordPwSer
1110ovprc2 5903 . . . . 5  |-  ( -.  R  e.  _V  ->  ( 1o ordPwSer  R )  =  (/) )
1211fveq1d 5543 . . . 4  |-  ( -.  R  e.  _V  ->  ( ( 1o ordPwSer  R ) `  (/) )  =  (
(/) `  (/) ) )
138, 9, 123eqtr4a 2354 . . 3  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
146, 13pm2.61i 156 . 2  |-  (PwSer1 `  R
)  =  ( ( 1o ordPwSer  R ) `  (/) )
151, 14eqtri 2316 1  |-  S  =  ( ( 1o ordPwSer  R ) `
 (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ` cfv 5271  (class class class)co 5874   1oc1o 6488   ordPwSer copws 16111  PwSer1cps1 16266
This theorem is referenced by:  psr1crng  16282  psr1assa  16283  psr1tos  16284  psr1bas2  16285  vr1cl2  16288  ply1lss  16291  ply1subrg  16292  psr1plusg  16316  psr1vsca  16317  psr1mulr  16318  psr1rng  16341  psr1lmod  16343  psr1sca  16344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-opsr 16122  df-psr1 16273
  Copyright terms: Public domain W3C validator