Theorem ply1val 22106

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 6822	. . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅))
3		ply1val.2	. . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅)
4	2, 3	eqtr4di 2784	. . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆)
5		oveq2 7354	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
6	5	fveq2d 6826	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
7	4, 6	oveq12d 7364	. . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
8		df-ply1 22094	. . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9		ovex 7379	. . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V
10	7, 8, 9	fvmpt 6929	. . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
11		fvprc 6814	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
12		ress0 17154	. . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
13	11, 12	eqtr4di 2784	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14		fvprc 6814	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
15	3, 14	eqtrid 2778	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
16	15	oveq1d 7361	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
17	13, 16	eqtr4d 2769	. . 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 2754	1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4280  ‘cfv 6481  (class class class)co 7346  1_oc1o 8378  Basecbs 17120   ↾_s cress 17141   mPoly cmpl 21843  PwSer₁cps1 22087  Poly₁cpl1 22089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-1cn 11064  ax-addcl 11066

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-nn 12126  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-ply1 22094

This theorem is referenced by:  ply1bas  22107  ply1basOLD  22108  ply1crng  22111  ply1assa  22112  ply1bascl  22116  ply1plusg  22136  ply1vsca  22137  ply1mulr  22138  ply1ring  22160  ply1lmod  22164  ply1sca  22165