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Theorem psr1val 16584
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 6089 . . . . 5  |-  ( r  =  R  ->  ( 1o ordPwSer  r )  =  ( 1o ordPwSer  R ) )
32fveq1d 5730 . . . 4  |-  ( r  =  R  ->  (
( 1o ordPwSer  r ) `  (/) )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
4 df-psr1 16576 . . . 4  |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `  (/) ) )
5 fvex 5742 . . . 4  |-  ( ( 1o ordPwSer  R ) `  (/) )  e. 
_V
63, 4, 5fvmpt 5806 . . 3  |-  ( R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
7 fv01 5763 . . . . 5  |-  ( (/) `  (/) )  =  (/)
87eqcomi 2440 . . . 4  |-  (/)  =  (
(/) `  (/) )
9 fvprc 5722 . . . 4  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
10 reldmopsr 16534 . . . . . 6  |-  Rel  dom ordPwSer
1110ovprc2 6110 . . . . 5  |-  ( -.  R  e.  _V  ->  ( 1o ordPwSer  R )  =  (/) )
1211fveq1d 5730 . . . 4  |-  ( -.  R  e.  _V  ->  ( ( 1o ordPwSer  R ) `  (/) )  =  (
(/) `  (/) ) )
138, 9, 123eqtr4a 2494 . . 3  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
146, 13pm2.61i 158 . 2  |-  (PwSer1 `  R
)  =  ( ( 1o ordPwSer  R ) `  (/) )
151, 14eqtri 2456 1  |-  S  =  ( ( 1o ordPwSer  R ) `
 (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ` cfv 5454  (class class class)co 6081   1oc1o 6717   ordPwSer copws 16414  PwSer1cps1 16569
This theorem is referenced by:  psr1crng  16585  psr1assa  16586  psr1tos  16587  psr1bas2  16588  vr1cl2  16591  ply1lss  16594  ply1subrg  16595  psr1plusg  16616  psr1vsca  16617  psr1mulr  16618  psr1rng  16641  psr1lmod  16643  psr1sca  16644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-opsr 16425  df-psr1 16576
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