Theorem ply1val 22120

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 6902	. . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅))
3		ply1val.2	. . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅)
4	2, 3	eqtr4di 2786	. . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆)
5		oveq2 7434	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
6	5	fveq2d 6906	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
7	4, 6	oveq12d 7444	. . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
8		df-ply1 22108	. . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9		ovex 7459	. . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V
10	7, 8, 9	fvmpt 7010	. . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
11		fvprc 6894	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
12		ress0 17231	. . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
13	11, 12	eqtr4di 2786	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14		fvprc 6894	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
15	3, 14	eqtrid 2780	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
16	15	oveq1d 7441	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
17	13, 16	eqtr4d 2771	. . 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 2756	1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ∅c0 4326  ‘cfv 6553  (class class class)co 7426  1_oc1o 8486  Basecbs 17187   ↾_s cress 17216   mPoly cmpl 21846  PwSer₁cps1 22101  Poly₁cpl1 22103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-1cn 11204  ax-addcl 11206

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-nn 12251  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-ply1 22108

This theorem is referenced by:  ply1bas  22121  ply1crng  22124  ply1assa  22125  ply1bascl  22129  ply1plusg  22149  ply1vsca  22150  ply1mulr  22151  ply1ring  22173  ply1lmod  22177  ply1sca  22178