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Theorem vr1val 20352
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 7156 . . . . 5 (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅))
32fveq1d 6665 . . . 4 (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅))
4 df-vr1 20341 . . . 4 var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
5 fvex 6676 . . . 4 ((1o mVar 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6761 . . 3 (𝑅 ∈ V → (var1𝑅) = ((1o mVar 𝑅)‘∅))
71, 6syl5eq 2866 . 2 (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
8 fvprc 6656 . . . 4 𝑅 ∈ V → (var1𝑅) = ∅)
9 0fv 6702 . . . 4 (∅‘∅) = ∅
108, 1, 93eqtr4g 2879 . . 3 𝑅 ∈ V → 𝑋 = (∅‘∅))
11 df-mvr 20129 . . . . . 6 mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
1211reldmmpo 7277 . . . . 5 Rel dom mVar
1312ovprc2 7188 . . . 4 𝑅 ∈ V → (1o mVar 𝑅) = ∅)
1413fveq1d 6665 . . 3 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅))
1510, 14eqtr4d 2857 . 2 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
167, 15pm2.61i 184 1 𝑋 = ((1o mVar 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  {crab 3140  Vcvv 3493  c0 4289  ifcif 4465  cmpt 5137  ccnv 5547  cima 5551  cfv 6348  (class class class)co 7148  1oc1o 8087  m cmap 8398  Fincfn 8501  0cc0 10529  1c1 10530  cn 11630  0cn0 11889  0gc0g 16705  1rcur 19243   mVar cmvr 20124  var1cv1 20336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-mvr 20129  df-vr1 20341
This theorem is referenced by:  vr1cl2  20353  vr1cl  20377  subrgvr1  20421  subrgvr1cl  20422  coe1tm  20433  ply1coe  20456  evl1var  20491  evls1var  20493
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