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Theorem psr1val 19917
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 6914 . . . . 5 (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
32fveq1d 6436 . . . 4 (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4 df-psr1 19911 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5 fvex 6447 . . . 4 ((1o ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6530 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
7 0fv 6474 . . . . 5 (∅‘∅) = ∅
87eqcomi 2835 . . . 4 ∅ = (∅‘∅)
9 fvprc 6427 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 19835 . . . . . 6 Rel dom ordPwSer
1110ovprc2 6945 . . . . 5 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
1211fveq1d 6436 . . . 4 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2888 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
146, 13pm2.61i 177 . 2 (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅)
151, 14eqtri 2850 1 𝑆 = ((1o ordPwSer 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1658  wcel 2166  Vcvv 3415  c0 4145  cfv 6124  (class class class)co 6906  1oc1o 7820   ordPwSer copws 19717  PwSer1cps1 19906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-opsr 19722  df-psr1 19911
This theorem is referenced by:  psr1crng  19918  psr1assa  19919  psr1tos  19920  psr1bas2  19921  vr1cl2  19924  ply1lss  19927  ply1subrg  19928  psr1plusg  19953  psr1vsca  19954  psr1mulr  19955  psr1ring  19978  psr1lmod  19980  psr1sca  19981
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