Theorem psr1val 19556

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 6658	. . . . 5 ⊢ (𝑟 = 𝑅 → (1𝑜 ordPwSer 𝑟) = (1𝑜 ordPwSer 𝑅))
3	2	fveq1d 6193	. . . 4 ⊢ (𝑟 = 𝑅 → ((1𝑜 ordPwSer 𝑟)‘∅) = ((1𝑜 ordPwSer 𝑅)‘∅))
4		df-psr1 19550	. . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
5		fvex 6201	. . . 4 ⊢ ((1𝑜 ordPwSer 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6282	. . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
7		0fv 6227	. . . . 5 ⊢ (∅‘∅) = ∅
8	7	eqcomi 2631	. . . 4 ⊢ ∅ = (∅‘∅)
9		fvprc 6185	. . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
10		reldmopsr 19473	. . . . . 6 ⊢ Rel dom ordPwSer
11	10	ovprc2 6685	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1𝑜 ordPwSer 𝑅) = ∅)
12	11	fveq1d 6193	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((1𝑜 ordPwSer 𝑅)‘∅) = (∅‘∅))
13	8, 9, 12	3eqtr4a 2682	. . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅))
14	6, 13	pm2.61i 176	. 2 ⊢ (PwSer1‘𝑅) = ((1𝑜 ordPwSer 𝑅)‘∅)
15	1, 14	eqtri 2644	1 ⊢ 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553   ordPwSer copws 19355  PwSer₁cps1 19545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-opsr 19360  df-psr1 19550

This theorem is referenced by:  psr1crng  19557  psr1assa  19558  psr1tos  19559  psr1bas2  19560  vr1cl2  19563  ply1lss  19566  ply1subrg  19567  psr1plusg  19592  psr1vsca  19593  psr1mulr  19594  psr1ring  19617  psr1lmod  19619  psr1sca  19620