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Theorem psr1val 19377
Description: Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypothesis
Ref Expression
psr1val.1 𝑆 = (PwSer1𝑅)
Assertion
Ref Expression
psr1val 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)

Proof of Theorem psr1val
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 psr1val.1 . 2 𝑆 = (PwSer1𝑅)
2 oveq2 6557 . . . . 5 (𝑟 = 𝑅 → (1𝑜 ordPwSer 𝑟) = (1𝑜 ordPwSer 𝑅))
32fveq1d 6105 . . . 4 (𝑟 = 𝑅 → ((1𝑜 ordPwSer 𝑟)‘∅) = ((1𝑜 ordPwSer 𝑅)‘∅))
4 df-psr1 19371 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
5 fvex 6113 . . . 4 ((1𝑜 ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6191 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
7 0fv 6137 . . . . 5 (∅‘∅) = ∅
87eqcomi 2619 . . . 4 ∅ = (∅‘∅)
9 fvprc 6097 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 19294 . . . . . 6 Rel dom ordPwSer
1110ovprc2 6583 . . . . 5 𝑅 ∈ V → (1𝑜 ordPwSer 𝑅) = ∅)
1211fveq1d 6105 . . . 4 𝑅 ∈ V → ((1𝑜 ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2670 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
146, 13pm2.61i 175 . 2 (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅)
151, 14eqtri 2632 1 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  cfv 5804  (class class class)co 6549  1𝑜c1o 7440   ordPwSer copws 19176  PwSer1cps1 19366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-opsr 19181  df-psr1 19371
This theorem is referenced by:  psr1crng  19378  psr1assa  19379  psr1tos  19380  psr1bas2  19381  vr1cl2  19384  ply1lss  19387  ply1subrg  19388  psr1plusg  19413  psr1vsca  19414  psr1mulr  19415  psr1ring  19438  psr1lmod  19440  psr1sca  19441
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