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Theorem psr1val 18036
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 6293 . . . . 5  |-  ( r  =  R  ->  ( 1o ordPwSer  r )  =  ( 1o ordPwSer  R ) )
32fveq1d 5868 . . . 4  |-  ( r  =  R  ->  (
( 1o ordPwSer  r ) `  (/) )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
4 df-psr1 18030 . . . 4  |- PwSer1  =  ( r  e.  _V  |->  ( ( 1o ordPwSer  r ) `  (/) ) )
5 fvex 5876 . . . 4  |-  ( ( 1o ordPwSer  R ) `  (/) )  e. 
_V
63, 4, 5fvmpt 5951 . . 3  |-  ( R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
7 0fv 5899 . . . . 5  |-  ( (/) `  (/) )  =  (/)
87eqcomi 2480 . . . 4  |-  (/)  =  (
(/) `  (/) )
9 fvprc 5860 . . . 4  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  (/) )
10 reldmopsr 17949 . . . . . 6  |-  Rel  dom ordPwSer
1110ovprc2 6314 . . . . 5  |-  ( -.  R  e.  _V  ->  ( 1o ordPwSer  R )  =  (/) )
1211fveq1d 5868 . . . 4  |-  ( -.  R  e.  _V  ->  ( ( 1o ordPwSer  R ) `  (/) )  =  (
(/) `  (/) ) )
138, 9, 123eqtr4a 2534 . . 3  |-  ( -.  R  e.  _V  ->  (PwSer1 `  R )  =  ( ( 1o ordPwSer  R ) `  (/) ) )
146, 13pm2.61i 164 . 2  |-  (PwSer1 `  R
)  =  ( ( 1o ordPwSer  R ) `  (/) )
151, 14eqtri 2496 1  |-  S  =  ( ( 1o ordPwSer  R ) `
 (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ` cfv 5588  (class class class)co 6285   1oc1o 7124   ordPwSer copws 17815  PwSer1cps1 18025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-opsr 17820  df-psr1 18030
This theorem is referenced by:  psr1crng  18037  psr1assa  18038  psr1tos  18039  psr1bas2  18040  vr1cl2  18043  ply1lss  18046  ply1subrg  18047  psr1plusg  18074  psr1vsca  18075  psr1mulr  18076  psr1rng  18099  psr1lmod  18101  psr1sca  18102
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