Theorem psr1val 19377

Description: Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypothesis

Ref	Expression
psr1val.1	⊢ 𝑆 = (PwSer1‘𝑅)

Assertion

Ref	Expression
psr1val	⊢ 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)

Proof of Theorem psr1val

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psr1val.1	. 2 ⊢ 𝑆 = (PwSer1‘𝑅)
2		oveq2 6557	. . . . 5 ⊢ (𝑟 = 𝑅 → (1𝑜 ordPwSer 𝑟) = (1𝑜 ordPwSer 𝑅))
3	2	fveq1d 6105	. . . 4 ⊢ (𝑟 = 𝑅 → ((1𝑜 ordPwSer 𝑟)‘∅) = ((1𝑜 ordPwSer 𝑅)‘∅))
4		df-psr1 19371	. . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
5		fvex 6113	. . . 4 ⊢ ((1𝑜 ordPwSer 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6191	. . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
7		0fv 6137	. . . . 5 ⊢ (∅‘∅) = ∅
8	7	eqcomi 2619	. . . 4 ⊢ ∅ = (∅‘∅)
9		fvprc 6097	. . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
10		reldmopsr 19294	. . . . . 6 ⊢ Rel dom ordPwSer
11	10	ovprc2 6583	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1𝑜 ordPwSer 𝑅) = ∅)
12	11	fveq1d 6105	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((1𝑜 ordPwSer 𝑅)‘∅) = (∅‘∅))
13	8, 9, 12	3eqtr4a 2670	. . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
14	6, 13	pm2.61i 175	. 2 ⊢ (PwSer1‘𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅)
15	1, 14	eqtri 2632	1 ⊢ 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ordPwSer copws 19176  PwSer₁cps1 19366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-opsr 19181  df-psr1 19371

This theorem is referenced by:  psr1crng  19378  psr1assa  19379  psr1tos  19380  psr1bas2  19381  vr1cl2  19384  ply1lss  19387  ply1subrg  19388  psr1plusg  19413  psr1vsca  19414  psr1mulr  19415  psr1ring  19438  psr1lmod  19440  psr1sca  19441