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Theorem psr1val 22187
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 7439 . . . . 5 (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
32fveq1d 6908 . . . 4 (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4 df-psr1 22181 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5 fvex 6919 . . . 4 ((1o ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 7016 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
7 0fv 6950 . . . . 5 (∅‘∅) = ∅
87eqcomi 2746 . . . 4 ∅ = (∅‘∅)
9 fvprc 6898 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 22063 . . . . . 6 Rel dom ordPwSer
1110ovprc2 7471 . . . . 5 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
1211fveq1d 6908 . . . 4 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2803 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
146, 13pm2.61i 182 . 2 (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅)
151, 14eqtri 2765 1 𝑆 = ((1o ordPwSer 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cfv 6561  (class class class)co 7431  1oc1o 8499   ordPwSer copws 21928  PwSer1cps1 22176
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-opsr 21933  df-psr1 22181
This theorem is referenced by:  psr1crng  22188  psr1assa  22189  psr1tos  22190  psr1bas2  22191  vr1cl2  22194  ply1lss  22198  ply1subrg  22199  psr1plusg  22222  psr1vsca  22223  psr1mulr  22224  psr1ring  22248  psr1lmod  22250  psr1sca  22251
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