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Theorem psr1val 18352
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 6304 . . . . 5  |-  ( r  =  R  ->  ( 1o ordPwSer  r )  =  ( 1o ordPwSer  R ) )
32fveq1d 5874 . . . 4  |-  ( r  =  R  ->  (
( 1o ordPwSer  r ) `  (/) )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
4 df-psr1 18346 . . . 4  |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `  (/) ) )
5 fvex 5882 . . . 4  |-  ( ( 1o ordPwSer  R ) `  (/) )  e. 
_V
63, 4, 5fvmpt 5956 . . 3  |-  ( R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
7 0fv 5905 . . . . 5  |-  ( (/) `  (/) )  =  (/)
87eqcomi 2470 . . . 4  |-  (/)  =  (
(/) `  (/) )
9 fvprc 5866 . . . 4  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
10 reldmopsr 18265 . . . . . 6  |-  Rel  dom ordPwSer
1110ovprc2 6328 . . . . 5  |-  ( -.  R  e.  _V  ->  ( 1o ordPwSer  R )  =  (/) )
1211fveq1d 5874 . . . 4  |-  ( -.  R  e.  _V  ->  ( ( 1o ordPwSer  R ) `  (/) )  =  (
(/) `  (/) ) )
138, 9, 123eqtr4a 2524 . . 3  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
146, 13pm2.61i 164 . 2  |-  (PwSer1 `  R
)  =  ( ( 1o ordPwSer  R ) `  (/) )
151, 14eqtri 2486 1  |-  S  =  ( ( 1o ordPwSer  R ) `
 (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ` cfv 5594  (class class class)co 6296   1oc1o 7141   ordPwSer copws 18131  PwSer1cps1 18341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-opsr 18136  df-psr1 18346
This theorem is referenced by:  psr1crng  18353  psr1assa  18354  psr1tos  18355  psr1bas2  18356  vr1cl2  18359  ply1lss  18362  ply1subrg  18363  psr1plusg  18390  psr1vsca  18391  psr1mulr  18392  psr1ring  18415  psr1lmod  18417  psr1sca  18418
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