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Theorem psr1val 19496
 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 6623 . . . . 5 (𝑟 = 𝑅 → (1𝑜 ordPwSer 𝑟) = (1𝑜 ordPwSer 𝑅))
32fveq1d 6160 . . . 4 (𝑟 = 𝑅 → ((1𝑜 ordPwSer 𝑟)‘∅) = ((1𝑜 ordPwSer 𝑅)‘∅))
4 df-psr1 19490 . . . 4 PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
5 fvex 6168 . . . 4 ((1𝑜 ordPwSer 𝑅)‘∅) ∈ V
63, 4, 5fvmpt 6249 . . 3 (𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
7 0fv 6194 . . . . 5 (∅‘∅) = ∅
87eqcomi 2630 . . . 4 ∅ = (∅‘∅)
9 fvprc 6152 . . . 4 𝑅 ∈ V → (PwSer1𝑅) = ∅)
10 reldmopsr 19413 . . . . . 6 Rel dom ordPwSer
1110ovprc2 6650 . . . . 5 𝑅 ∈ V → (1𝑜 ordPwSer 𝑅) = ∅)
1211fveq1d 6160 . . . 4 𝑅 ∈ V → ((1𝑜 ordPwSer 𝑅)‘∅) = (∅‘∅))
138, 9, 123eqtr4a 2681 . . 3 𝑅 ∈ V → (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
146, 13pm2.61i 176 . 2 (PwSer1𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅)
151, 14eqtri 2643 1 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1480   ∈ wcel 1987  Vcvv 3190  ∅c0 3897  ‘cfv 5857  (class class class)co 6615  1𝑜c1o 7513   ordPwSer copws 19295  PwSer1cps1 19485 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-opsr 19300  df-psr1 19490 This theorem is referenced by:  psr1crng  19497  psr1assa  19498  psr1tos  19499  psr1bas2  19500  vr1cl2  19503  ply1lss  19506  ply1subrg  19507  psr1plusg  19532  psr1vsca  19533  psr1mulr  19534  psr1ring  19557  psr1lmod  19559  psr1sca  19560
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