Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  vr1val Structured version   Unicode version

Theorem vr1val 18358
 Description: The value of the generator of the power series algebra (the in ). Since all univariate polynomial rings over a fixed base ring 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 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 var1
Assertion
Ref Expression
vr1val mVar

Proof of Theorem vr1val
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vr1val.1 . . 3 var1
2 oveq2 6304 . . . . 5 mVar mVar
32fveq1d 5874 . . . 4 mVar mVar
4 df-vr1 18347 . . . 4 var1 mVar
5 fvex 5882 . . . 4 mVar
63, 4, 5fvmpt 5956 . . 3 var1 mVar
71, 6syl5eq 2510 . 2 mVar
8 fvprc 5866 . . . 4 var1
9 0fv 5905 . . . 4
108, 1, 93eqtr4g 2523 . . 3
11 df-mvr 18133 . . . . . 6 mVar
1211reldmmpt2 6412 . . . . 5 mVar
1312ovprc2 6328 . . . 4 mVar
1413fveq1d 5874 . . 3 mVar
1510, 14eqtr4d 2501 . 2 mVar
167, 15pm2.61i 164 1 mVar
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1395   wcel 1819  crab 2811  cvv 3109  c0 3793  cif 3944   cmpt 4515  ccnv 5007  cima 5011  cfv 5594  (class class class)co 6296  c1o 7141   cmap 7438  cfn 7535  cc0 9509  c1 9510  cn 10556  cn0 10816  c0g 14857  cur 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
 Copyright terms: Public domain W3C validator