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Theorem psr1val 19750
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 6813 . . . . 5 (𝑟 = 𝑅 → (1𝑜 ordPwSer 𝑟) = (1𝑜 ordPwSer 𝑅))
32fveq1d 6346 . . . 4 (𝑟 = 𝑅 → ((1𝑜 ordPwSer 𝑟)‘∅) = ((1𝑜 ordPwSer 𝑅)‘∅))
4 df-psr1 19744 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
5 fvex 6354 . . . 4 ((1𝑜 ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6436 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
7 0fv 6380 . . . . 5 (∅‘∅) = ∅
87eqcomi 2761 . . . 4 ∅ = (∅‘∅)
9 fvprc 6338 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 19667 . . . . . 6 Rel dom ordPwSer
1110ovprc2 6840 . . . . 5 𝑅 ∈ V → (1𝑜 ordPwSer 𝑅) = ∅)
1211fveq1d 6346 . . . 4 𝑅 ∈ V → ((1𝑜 ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2812 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
146, 13pm2.61i 176 . 2 (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅)
151, 14eqtri 2774 1 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1624  wcel 2131  Vcvv 3332  c0 4050  cfv 6041  (class class class)co 6805  1𝑜c1o 7714   ordPwSer copws 19549  PwSer1cps1 19739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-opsr 19554  df-psr1 19744
This theorem is referenced by:  psr1crng  19751  psr1assa  19752  psr1tos  19753  psr1bas2  19754  vr1cl2  19757  ply1lss  19760  ply1subrg  19761  psr1plusg  19786  psr1vsca  19787  psr1mulr  19788  psr1ring  19811  psr1lmod  19813  psr1sca  19814
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