Theorem psr1val 21710

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 7417	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
3	2	fveq1d 6894	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4		df-psr1 21704	. . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5		fvex 6905	. . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6999	. . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
7		0fv 6936	. . . . 5 ⊢ (∅‘∅) = ∅
8	7	eqcomi 2742	. . . 4 ⊢ ∅ = (∅‘∅)
9		fvprc 6884	. . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
10		reldmopsr 21600	. . . . . 6 ⊢ Rel dom ordPwSer
11	10	ovprc2 7449	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
12	11	fveq1d 6894	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
13	8, 9, 12	3eqtr4a 2799	. . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
14	6, 13	pm2.61i 182	. 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)
15	1, 14	eqtri 2761	1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ∅c0 4323  ‘cfv 6544  (class class class)co 7409  1_oc1o 8459   ordPwSer copws 21461  PwSer₁cps1 21699

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-opsr 21466  df-psr1 21704

This theorem is referenced by:  psr1crng  21711  psr1assa  21712  psr1tos  21713  psr1bas2  21714  vr1cl2  21717  ply1lss  21720  ply1subrg  21721  psr1plusg  21744  psr1vsca  21745  psr1mulr  21746  psr1ring  21769  psr1lmod  21771  psr1sca  21772