Theorem ply1val 20365

Description: The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypotheses

Ref	Expression
ply1val.1	⊢ 𝑃 = (Poly1‘𝑅)
ply1val.2	⊢ 𝑆 = (PwSer1‘𝑅)

Assertion

Ref	Expression
ply1val	⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))

Proof of Theorem ply1val

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1val.1	. 2 ⊢ 𝑃 = (Poly1‘𝑅)
2		fveq2 6673	. . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅))
3		ply1val.2	. . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅)
4	2, 3	syl6eqr 2877	. . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆)
5		oveq2 7167	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
6	5	fveq2d 6677	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
7	4, 6	oveq12d 7177	. . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
8		df-ply1 20353	. . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9		ovex 7192	. . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V
10	7, 8, 9	fvmpt 6771	. . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
11		fvprc 6666	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
12		ress0 16561	. . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
13	11, 12	syl6eqr 2877	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14		fvprc 6666	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
15	3, 14	syl5eq 2871	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
16	15	oveq1d 7174	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
17	13, 16	eqtr4d 2862	. . 3 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
18	10, 17	pm2.61i 184	. 2 ⊢ (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))
19	1, 18	eqtri 2847	1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  Vcvv 3497  ∅c0 4294  ‘cfv 6358  (class class class)co 7159  1_oc1o 8098  Basecbs 16486   ↾_s cress 16487   mPoly cmpl 20136  PwSer₁cps1 20346  Poly₁cpl1 20348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-slot 16490  df-base 16492  df-ress 16494  df-ply1 20353

This theorem is referenced by:  ply1bas  20366  ply1crng  20369  ply1assa  20370  ply1bascl  20374  ply1plusg  20396  ply1vsca  20397  ply1mulr  20398  ply1ring  20419  ply1lmod  20423  ply1sca  20424