Theorem psr1val 22121

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 7413	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
3	2	fveq1d 6878	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4		df-psr1 22115	. . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5		fvex 6889	. . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6986	. . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
7		0fv 6920	. . . . 5 ⊢ (∅‘∅) = ∅
8	7	eqcomi 2744	. . . 4 ⊢ ∅ = (∅‘∅)
9		fvprc 6868	. . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
10		reldmopsr 22003	. . . . . 6 ⊢ Rel dom ordPwSer
11	10	ovprc2 7445	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
12	11	fveq1d 6878	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
13	8, 9, 12	3eqtr4a 2796	. . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
14	6, 13	pm2.61i 182	. 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)
15	1, 14	eqtri 2758	1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  ‘cfv 6531  (class class class)co 7405  1_oc1o 8473   ordPwSer copws 21868  PwSer₁cps1 22110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-opsr 21873  df-psr1 22115

This theorem is referenced by:  psr1crng  22122  psr1assa  22123  psr1tos  22124  psr1bas2  22125  vr1cl2  22128  ply1lss  22132  ply1subrg  22133  psr1plusg  22156  psr1vsca  22157  psr1mulr  22158  psr1ring  22182  psr1lmod  22184  psr1sca  22185