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Theorem psr1val 22105
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 7428 . . . . 5 (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
32fveq1d 6899 . . . 4 (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4 df-psr1 22099 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5 fvex 6910 . . . 4 ((1o ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 7005 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
7 0fv 6941 . . . . 5 (∅‘∅) = ∅
87eqcomi 2737 . . . 4 ∅ = (∅‘∅)
9 fvprc 6889 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 21983 . . . . . 6 Rel dom ordPwSer
1110ovprc2 7460 . . . . 5 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
1211fveq1d 6899 . . . 4 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2794 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅))
146, 13pm2.61i 182 . 2 (PwSer1𝑅) = ((1o ordPwSer 𝑅)‘∅)
151, 14eqtri 2756 1 𝑆 = ((1o ordPwSer 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  Vcvv 3471  c0 4323  cfv 6548  (class class class)co 7420  1oc1o 8480   ordPwSer copws 21841  PwSer1cps1 22094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-opsr 21846  df-psr1 22099
This theorem is referenced by:  psr1crng  22106  psr1assa  22107  psr1tos  22108  psr1bas2  22109  vr1cl2  22112  ply1lss  22115  ply1subrg  22116  psr1plusg  22139  psr1vsca  22140  psr1mulr  22141  psr1ring  22165  psr1lmod  22167  psr1sca  22168
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