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Theorem psr1val 19829
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 6850 . . . . 5 (𝑟 = 𝑅 → (1𝑜 ordPwSer 𝑟) = (1𝑜 ordPwSer 𝑅))
32fveq1d 6377 . . . 4 (𝑟 = 𝑅 → ((1𝑜 ordPwSer 𝑟)‘∅) = ((1𝑜 ordPwSer 𝑅)‘∅))
4 df-psr1 19823 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
5 fvex 6388 . . . 4 ((1𝑜 ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6471 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
7 0fv 6415 . . . . 5 (∅‘∅) = ∅
87eqcomi 2774 . . . 4 ∅ = (∅‘∅)
9 fvprc 6368 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 19747 . . . . . 6 Rel dom ordPwSer
1110ovprc2 6881 . . . . 5 𝑅 ∈ V → (1𝑜 ordPwSer 𝑅) = ∅)
1211fveq1d 6377 . . . 4 𝑅 ∈ V → ((1𝑜 ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2825 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
146, 13pm2.61i 176 . 2 (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅)
151, 14eqtri 2787 1 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  cfv 6068  (class class class)co 6842  1𝑜c1o 7757   ordPwSer copws 19629  PwSer1cps1 19818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-opsr 19634  df-psr1 19823
This theorem is referenced by:  psr1crng  19830  psr1assa  19831  psr1tos  19832  psr1bas2  19833  vr1cl2  19836  ply1lss  19839  ply1subrg  19840  psr1plusg  19865  psr1vsca  19866  psr1mulr  19867  psr1ring  19890  psr1lmod  19892  psr1sca  19893
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