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Theorem vr1val 19776
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 6800 . . . . 5 (𝑟 = 𝑅 → (1𝑜 mVar 𝑟) = (1𝑜 mVar 𝑅))
32fveq1d 6334 . . . 4 (𝑟 = 𝑅 → ((1𝑜 mVar 𝑟)‘∅) = ((1𝑜 mVar 𝑅)‘∅))
4 df-vr1 19765 . . . 4 var1 = (𝑟 ∈ V ↦ ((1𝑜 mVar 𝑟)‘∅))
5 fvex 6342 . . . 4 ((1𝑜 mVar 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6424 . . 3 (𝑅 ∈ V → (var1𝑅) = ((1𝑜 mVar 𝑅)‘∅))
71, 6syl5eq 2816 . 2 (𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
8 fvprc 6326 . . . 4 𝑅 ∈ V → (var1𝑅) = ∅)
9 0fv 6368 . . . 4 (∅‘∅) = ∅
108, 1, 93eqtr4g 2829 . . 3 𝑅 ∈ V → 𝑋 = (∅‘∅))
11 df-mvr 19571 . . . . . 6 mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
1211reldmmpt2 6917 . . . . 5 Rel dom mVar
1312ovprc2 6829 . . . 4 𝑅 ∈ V → (1𝑜 mVar 𝑅) = ∅)
1413fveq1d 6334 . . 3 𝑅 ∈ V → ((1𝑜 mVar 𝑅)‘∅) = (∅‘∅))
1510, 14eqtr4d 2807 . 2 𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
167, 15pm2.61i 176 1 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1630  wcel 2144  {crab 3064  Vcvv 3349  c0 4061  ifcif 4223  cmpt 4861  ccnv 5248  cima 5252  cfv 6031  (class class class)co 6792  1𝑜c1o 7705  𝑚 cmap 8008  Fincfn 8108  0cc0 10137  1c1 10138  cn 11221  0cn0 11493  0gc0g 16307  1rcur 18708   mVar cmvr 19566  var1cv1 19760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-mvr 19571  df-vr1 19765
This theorem is referenced by:  vr1cl2  19777  vr1cl  19801  subrgvr1  19845  subrgvr1cl  19846  coe1tm  19857  ply1coe  19880  evl1var  19914  evls1var  19916
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