Theorem ply1val 19385

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 6103	. . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅))
3		ply1val.2	. . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅)
4	2, 3	syl6eqr 2662	. . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆)
5		oveq2 6557	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1𝑜 mPoly 𝑟) = (1𝑜 mPoly 𝑅))
6	5	fveq2d 6107	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1𝑜 mPoly 𝑟)) = (Base‘(1𝑜 mPoly 𝑅)))
7	4, 6	oveq12d 6567	. . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))))
8		df-ply1 19373	. . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))))
9		ovex 6577	. . . 4 ⊢ (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) ∈ V
10	7, 8, 9	fvmpt 6191	. . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))))
11		fvprc 6097	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
12		ress0 15761	. . . . 5 ⊢ (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))) = ∅
13	11, 12	syl6eqr 2662	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))))
14		fvprc 6097	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
15	3, 14	syl5eq 2656	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
16	15	oveq1d 6564	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))))
17	13, 16	eqtr4d 2647	. . 3 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))))
18	10, 17	pm2.61i 175	. 2 ⊢ (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))
19	1, 18	eqtri 2632	1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅)))

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440  Basecbs 15695   ↾_s cress 15696   mPoly cmpl 19174  PwSer₁cps1 19366  Poly₁cpl1 19368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-slot 15699  df-base 15700  df-ress 15702  df-ply1 19373

This theorem is referenced by:  ply1bas  19386  ply1crng  19389  ply1assa  19390  ply1bascl  19394  ply1plusg  19416  ply1vsca  19417  ply1mulr  19418  ply1ring  19439  ply1lmod  19443  ply1sca  19444