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

Theorem vr1val 22193
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 = (var1𝑅)
Assertion
Ref Expression
vr1val 𝑋 = ((1o mVar 𝑅)‘∅)

Proof of Theorem vr1val
Dummy variables 𝑓 𝑖 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vr1val.1 . . 3 𝑋 = (var1𝑅)
2 oveq2 7439 . . . . 5 (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅))
32fveq1d 6908 . . . 4 (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅))
4 df-vr1 22182 . . . 4 var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
5 fvex 6919 . . . 4 ((1o mVar 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 7016 . . 3 (𝑅 ∈ V → (var1𝑅) = ((1o mVar 𝑅)‘∅))
71, 6eqtrid 2789 . 2 (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
8 fvprc 6898 . . . 4 𝑅 ∈ V → (var1𝑅) = ∅)
9 0fv 6950 . . . 4 (∅‘∅) = ∅
108, 1, 93eqtr4g 2802 . . 3 𝑅 ∈ V → 𝑋 = (∅‘∅))
11 df-mvr 21930 . . . . . 6 mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
1211reldmmpo 7567 . . . . 5 Rel dom mVar
1312ovprc2 7471 . . . 4 𝑅 ∈ V → (1o mVar 𝑅) = ∅)
1413fveq1d 6908 . . 3 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅))
1510, 14eqtr4d 2780 . 2 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
167, 15pm2.61i 182 1 𝑋 = ((1o mVar 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  c0 4333  ifcif 4525  cmpt 5225  ccnv 5684  cima 5688  cfv 6561  (class class class)co 7431  1oc1o 8499  m cmap 8866  Fincfn 8985  0cc0 11155  1c1 11156  cn 12266  0cn0 12526  0gc0g 17484  1rcur 20178   mVar cmvr 21925  var1cv1 22177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mvr 21930  df-vr1 22182
This theorem is referenced by:  vr1cl2  22194  vr1cl  22219  subrgvr1  22264  subrgvr1cl  22265  coe1tm  22276  ply1coe  22302  evl1var  22340  evls1var  22342  rhmply1vr1  22391
  Copyright terms: Public domain W3C validator