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Theorem psr1val 16265
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 5866 . . . . 5  |-  ( r  =  R  ->  ( 1o ordPwSer  r )  =  ( 1o ordPwSer  R ) )
32fveq1d 5527 . . . 4  |-  ( r  =  R  ->  (
( 1o ordPwSer  r ) `  (/) )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
4 df-psr1 16257 . . . 4  |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `  (/) ) )
5 fvex 5539 . . . 4  |-  ( ( 1o ordPwSer  R ) `  (/) )  e. 
_V
63, 4, 5fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
7 fv01 5559 . . . . 5  |-  ( (/) `  (/) )  =  (/)
87eqcomi 2287 . . . 4  |-  (/)  =  (
(/) `  (/) )
9 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
10 reldmopsr 16215 . . . . . 6  |-  Rel  dom ordPwSer
1110ovprc2 5887 . . . . 5  |-  ( -.  R  e.  _V  ->  ( 1o ordPwSer  R )  =  (/) )
1211fveq1d 5527 . . . 4  |-  ( -.  R  e.  _V  ->  ( ( 1o ordPwSer  R ) `  (/) )  =  (
(/) `  (/) ) )
138, 9, 123eqtr4a 2341 . . 3  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
146, 13pm2.61i 156 . 2  |-  (PwSer1 `  R
)  =  ( ( 1o ordPwSer  R ) `  (/) )
151, 14eqtri 2303 1  |-  S  =  ( ( 1o ordPwSer  R ) `
 (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ` cfv 5255  (class class class)co 5858   1oc1o 6472   ordPwSer copws 16095  PwSer1cps1 16250
This theorem is referenced by:  psr1crng  16266  psr1assa  16267  psr1tos  16268  psr1bas2  16269  vr1cl2  16272  ply1lss  16275  ply1subrg  16276  psr1plusg  16300  psr1vsca  16301  psr1mulr  16302  psr1rng  16325  psr1lmod  16327  psr1sca  16328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-opsr 16106  df-psr1 16257
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