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Theorem vr1val 18358
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 6304 . . . . 5  |-  ( r  =  R  ->  ( 1o mVar  r )  =  ( 1o mVar  R ) )
32fveq1d 5874 . . . 4  |-  ( r  =  R  ->  (
( 1o mVar  r ) `  (/) )  =  ( ( 1o mVar  R ) `
 (/) ) )
4 df-vr1 18347 . . . 4  |- var1  =  (
r  e.  _V  |->  ( ( 1o mVar  r ) `
 (/) ) )
5 fvex 5882 . . . 4  |-  ( ( 1o mVar  R ) `  (/) )  e.  _V
63, 4, 5fvmpt 5956 . . 3  |-  ( R  e.  _V  ->  (var1 `  R )  =  ( ( 1o mVar  R ) `
 (/) ) )
71, 6syl5eq 2510 . 2  |-  ( R  e.  _V  ->  X  =  ( ( 1o mVar  R ) `  (/) ) )
8 fvprc 5866 . . . 4  |-  ( -.  R  e.  _V  ->  (var1 `  R )  =  (/) )
9 0fv 5905 . . . 4  |-  ( (/) `  (/) )  =  (/)
108, 1, 93eqtr4g 2523 . . 3  |-  ( -.  R  e.  _V  ->  X  =  ( (/) `  (/) ) )
11 df-mvr 18133 . . . . . 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 6412 . . . . 5  |-  Rel  dom mVar
1312ovprc2 6328 . . . 4  |-  ( -.  R  e.  _V  ->  ( 1o mVar  R )  =  (/) )
1413fveq1d 5874 . . 3  |-  ( -.  R  e.  _V  ->  ( ( 1o mVar  R ) `
 (/) )  =  (
(/) `  (/) ) )
1510, 14eqtr4d 2501 . 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   (/)c0 3793   ifcif 3944    |-> cmpt 4515   `'ccnv 5007   "cima 5011   ` cfv 5594  (class class class)co 6296   1oc1o 7141    ^m cmap 7438   Fincfn 7535   0cc0 9509   1c1 9510   NNcn 10556   NN0cn0 10816   0gc0g 14857   1rcur 17280   mVar cmvr 18128  var1cv1 18342
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-mvr 18133  df-vr1 18347
This theorem is referenced by:  vr1cl2  18359  vr1cl  18385  subrgvr1  18429  subrgvr1cl  18430  coe1tm  18441  ply1coe  18464  ply1coeOLD  18465  evl1var  18499  evls1var  18501
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