Theorem ply1val 22524

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 6893	. . . . . 6 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = (PwSer1‘𝑅))
3		ply1val.2	. . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅)
4	2, 3	eqtr4di 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → (PwSer1‘𝑟) = 𝑆)
5		oveq2 7435	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
6	5	fveq2d 6897	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
7	4, 6	oveq12d 7445	. . . 4 ⊢ (𝑟 = 𝑅 → ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
8		df-ply1 22512	. . . 4 ⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9		ovex 7460	. . . 4 ⊢ (𝑆 ↾s (Base‘(1o mPoly 𝑅))) ∈ V
10	7, 8, 9	fvmpt 7001	. . 3 ⊢ (𝑅 ∈ V → (Poly1‘𝑅) = (𝑆 ↾s (Base‘(1o mPoly 𝑅))))
11		fvprc 6885	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
12		ress0 17433	. . . . 5 ⊢ (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
13	11, 12	eqtr4di 2816	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14		fvprc 6885	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅)
15	3, 14	eqtrid 2810	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
16	15	oveq1d 7442	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑆 ↾s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
17	13, 16	eqtr4d 2801	. . 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 2786	1 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ∅c0 4280  ‘cfv 6546  (class class class)co 7427  1_oc1o 8480  Basecbs 17399   ↾_s cress 17420   mPoly cmpl 22226  PwSer₁cps1 22505  Poly₁cpl1 22507

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-1cn 11268  ax-addcl 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-nn 12346  df-slot 17372  df-ndx 17384  df-base 17400  df-ress 17421  df-ply1 22512

This theorem is used by:  ply1bas  22525  ply1crng  22528  ply1assa  22529  ply1bascl  22533  ply1plusg  22553  ply1vsca  22554  ply1mulr  22555  ply1ring  22577  ply1lmod  22581  ply1sca  22582