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Theorem psr1val 21709
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 7416 . . . . 5 (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
32fveq1d 6893 . . . 4 (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4 df-psr1 21703 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5 fvex 6904 . . . 4 ((1o ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6998 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
7 0fv 6935 . . . . 5 (∅‘∅) = ∅
87eqcomi 2741 . . . 4 ∅ = (∅‘∅)
9 fvprc 6883 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 21599 . . . . . 6 Rel dom ordPwSer
1110ovprc2 7448 . . . . 5 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
1211fveq1d 6893 . . . 4 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2798 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
146, 13pm2.61i 182 . 2 (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅)
151, 14eqtri 2760 1 𝑆 = ((1o ordPwSer 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  Vcvv 3474  c0 4322  cfv 6543  (class class class)co 7408  1oc1o 8458   ordPwSer copws 21460  PwSer1cps1 21698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-opsr 21465  df-psr1 21703
This theorem is referenced by:  psr1crng  21710  psr1assa  21711  psr1tos  21712  psr1bas2  21713  vr1cl2  21716  ply1lss  21719  ply1subrg  21720  psr1plusg  21743  psr1vsca  21744  psr1mulr  21745  psr1ring  21768  psr1lmod  21770  psr1sca  21771
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