Theorem ply1val 22078

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 6858	. . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅))
3		ply1val.2	. . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅)
4	2, 3	eqtr4di 2782	. . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆)
5		oveq2 7395	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
6	5	fveq2d 6862	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
7	4, 6	oveq12d 7405	. . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
8		df-ply1 22066	. . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9		ovex 7420	. . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V
10	7, 8, 9	fvmpt 6968	. . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
11		fvprc 6850	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
12		ress0 17213	. . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
13	11, 12	eqtr4di 2782	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14		fvprc 6850	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
15	3, 14	eqtrid 2776	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
16	15	oveq1d 7402	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
17	13, 16	eqtr4d 2767	. . 3 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
18	10, 17	pm2.61i 182	. 2 ⊢ (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))
19	1, 18	eqtri 2752	1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ∅c0 4296  ‘cfv 6511  (class class class)co 7387  1_oc1o 8427  Basecbs 17179   ↾_s cress 17200   mPoly cmpl 21815  PwSer₁cps1 22059  Poly₁cpl1 22061

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-nn 12187  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-ply1 22066

This theorem is referenced by:  ply1bas  22079  ply1basOLD  22080  ply1crng  22083  ply1assa  22084  ply1bascl  22088  ply1plusg  22108  ply1vsca  22109  ply1mulr  22110  ply1ring  22132  ply1lmod  22136  ply1sca  22137