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Theorem vr1val 17999
Description: The value of the generator of the power series algebra (the  X in  R [ [ X ] ]). Since all univariate polynomial rings over a fixed base ring  R are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and  1o  =  { (/) } is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
Hypothesis
Ref Expression
vr1val.1  |-  X  =  (var1 `  R )
Assertion
Ref Expression
vr1val  |-  X  =  ( ( 1o mVar  R
) `  (/) )

Proof of Theorem vr1val
Dummy variables  f  h  i  r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vr1val.1 . . 3  |-  X  =  (var1 `  R )
2 oveq2 6290 . . . . 5  |-  ( r  =  R  ->  ( 1o mVar  r )  =  ( 1o mVar  R ) )
32fveq1d 5866 . . . 4  |-  ( r  =  R  ->  (
( 1o mVar  r ) `  (/) )  =  ( ( 1o mVar  R ) `
 (/) ) )
4 df-vr1 17988 . . . 4  |- var1  =  (
r  e.  _V  |->  ( ( 1o mVar  r ) `
 (/) ) )
5 fvex 5874 . . . 4  |-  ( ( 1o mVar  R ) `  (/) )  e.  _V
63, 4, 5fvmpt 5948 . . 3  |-  ( R  e.  _V  ->  (var1 `  R )  =  ( ( 1o mVar  R ) `
 (/) ) )
71, 6syl5eq 2520 . 2  |-  ( R  e.  _V  ->  X  =  ( ( 1o mVar  R ) `  (/) ) )
8 fvprc 5858 . . . 4  |-  ( -.  R  e.  _V  ->  (var1 `  R )  =  (/) )
9 0fv 5897 . . . 4  |-  ( (/) `  (/) )  =  (/)
108, 1, 93eqtr4g 2533 . . 3  |-  ( -.  R  e.  _V  ->  X  =  ( (/) `  (/) ) )
11 df-mvr 17774 . . . . . 6  |- mVar  =  ( i  e.  _V , 
r  e.  _V  |->  ( x  e.  i  |->  ( f  e.  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  |->  if ( f  =  ( y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) ) ,  ( 1r `  r ) ,  ( 0g `  r ) ) ) ) )
1211reldmmpt2 6395 . . . . 5  |-  Rel  dom mVar
1312ovprc2 6311 . . . 4  |-  ( -.  R  e.  _V  ->  ( 1o mVar  R )  =  (/) )
1413fveq1d 5866 . . 3  |-  ( -.  R  e.  _V  ->  ( ( 1o mVar  R ) `
 (/) )  =  (
(/) `  (/) ) )
1510, 14eqtr4d 2511 . 2  |-  ( -.  R  e.  _V  ->  X  =  ( ( 1o mVar  R ) `  (/) ) )
167, 15pm2.61i 164 1  |-  X  =  ( ( 1o mVar  R
) `  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   (/)c0 3785   ifcif 3939    |-> cmpt 4505   `'ccnv 4998   "cima 5002   ` cfv 5586  (class class class)co 6282   1oc1o 7120    ^m cmap 7417   Fincfn 7513   0cc0 9488   1c1 9489   NNcn 10532   NN0cn0 10791   0gc0g 14688   1rcur 16940   mVar cmvr 17769  var1cv1 17983
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 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-mvr 17774  df-vr1 17988
This theorem is referenced by:  vr1cl2  18000  vr1cl  18026  subrgvr1  18070  subrgvr1cl  18071  coe1tm  18082  ply1coe  18105  ply1coeOLD  18106  evl1var  18140  evls1var  18142
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