Theorem psr1val 22149

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 7375	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅))
3	2	fveq1d 6842	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅))
4		df-psr1 22143	. . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
5		fvex 6853	. . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6947	. . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
7		0fv 6881	. . . . 5 ⊢ (∅‘∅) = ∅
8	7	eqcomi 2745	. . . 4 ⊢ ∅ = (∅‘∅)
9		fvprc 6832	. . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
10		reldmopsr 22023	. . . . . 6 ⊢ Rel dom ordPwSer
11	10	ovprc2 7407	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅)
12	11	fveq1d 6842	. . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅))
13	8, 9, 12	3eqtr4a 2797	. . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅))
14	6, 13	pm2.61i 182	. 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)
15	1, 14	eqtri 2759	1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273  ‘cfv 6498  (class class class)co 7367  1_oc1o 8398   ordPwSer copws 21888  PwSer₁cps1 22138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-opsr 21893  df-psr1 22143

This theorem is referenced by:  psr1crng  22150  psr1assa  22151  psr1tos  22152  psr1bas2  22153  vr1cl2  22156  ply1lss  22160  ply1subrg  22161  psr1plusg  22184  psr1vsca  22185  psr1mulr  22186  psr1ring  22210  psr1lmod  22212  psr1sca  22213