Theorem psr1val 22203

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 7439	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
3	2	fveq1d 6909	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4		df-psr1 22197	. . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5		fvex 6920	. . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 7016	. . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
7		0fv 6951	. . . . 5 ⊢ (∅‘∅) = ∅
8	7	eqcomi 2744	. . . 4 ⊢ ∅ = (∅‘∅)
9		fvprc 6899	. . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
10		reldmopsr 22081	. . . . . 6 ⊢ Rel dom ordPwSer
11	10	ovprc2 7471	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
12	11	fveq1d 6909	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
13	8, 9, 12	3eqtr4a 2801	. . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
14	6, 13	pm2.61i 182	. 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)
15	1, 14	eqtri 2763	1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  ‘cfv 6563  (class class class)co 7431  1_oc1o 8498   ordPwSer copws 21946  PwSer₁cps1 22192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-opsr 21951  df-psr1 22197

This theorem is referenced by:  psr1crng  22204  psr1assa  22205  psr1tos  22206  psr1bas2  22207  vr1cl2  22210  ply1lss  22214  ply1subrg  22215  psr1plusg  22238  psr1vsca  22239  psr1mulr  22240  psr1ring  22264  psr1lmod  22266  psr1sca  22267