Theorem psr1val 21357

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 7283	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
3	2	fveq1d 6776	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4		df-psr1 21351	. . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5		fvex 6787	. . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6875	. . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
7		0fv 6813	. . . . 5 ⊢ (∅‘∅) = ∅
8	7	eqcomi 2747	. . . 4 ⊢ ∅ = (∅‘∅)
9		fvprc 6766	. . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
10		reldmopsr 21246	. . . . . 6 ⊢ Rel dom ordPwSer
11	10	ovprc2 7315	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
12	11	fveq1d 6776	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
13	8, 9, 12	3eqtr4a 2804	. . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
14	6, 13	pm2.61i 182	. 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)
15	1, 14	eqtri 2766	1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290   ordPwSer copws 21111  PwSer₁cps1 21346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-opsr 21116  df-psr1 21351

This theorem is referenced by:  psr1crng  21358  psr1assa  21359  psr1tos  21360  psr1bas2  21361  vr1cl2  21364  ply1lss  21367  ply1subrg  21368  psr1plusg  21393  psr1vsca  21394  psr1mulr  21395  psr1ring  21418  psr1lmod  21420  psr1sca  21421