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Theorem ply1val 19778
Description: The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypotheses
Ref Expression
ply1val.1 𝑃 = (Poly1𝑅)
ply1val.2 𝑆 = (PwSer1𝑅)
Assertion
Ref Expression
ply1val 𝑃 = (𝑆s (Base‘(1𝑜 mPoly 𝑅)))

Proof of Theorem ply1val
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 ply1val.1 . 2 𝑃 = (Poly1𝑅)
2 fveq2 6332 . . . . . 6 (𝑟 = 𝑅 → (PwSer1𝑟) = (PwSer1𝑅))
3 ply1val.2 . . . . . 6 𝑆 = (PwSer1𝑅)
42, 3syl6eqr 2822 . . . . 5 (𝑟 = 𝑅 → (PwSer1𝑟) = 𝑆)
5 oveq2 6800 . . . . . 6 (𝑟 = 𝑅 → (1𝑜 mPoly 𝑟) = (1𝑜 mPoly 𝑅))
65fveq2d 6336 . . . . 5 (𝑟 = 𝑅 → (Base‘(1𝑜 mPoly 𝑟)) = (Base‘(1𝑜 mPoly 𝑅)))
74, 6oveq12d 6810 . . . 4 (𝑟 = 𝑅 → ((PwSer1𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))) = (𝑆s (Base‘(1𝑜 mPoly 𝑅))))
8 df-ply1 19766 . . . 4 Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))))
9 ovex 6822 . . . 4 (𝑆s (Base‘(1𝑜 mPoly 𝑅))) ∈ V
107, 8, 9fvmpt 6424 . . 3 (𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1𝑜 mPoly 𝑅))))
11 fvprc 6326 . . . . 5 𝑅 ∈ V → (Poly1𝑅) = ∅)
12 ress0 16140 . . . . 5 (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))) = ∅
1311, 12syl6eqr 2822 . . . 4 𝑅 ∈ V → (Poly1𝑅) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))))
14 fvprc 6326 . . . . . 6 𝑅 ∈ V → (PwSer1𝑅) = ∅)
153, 14syl5eq 2816 . . . . 5 𝑅 ∈ V → 𝑆 = ∅)
1615oveq1d 6807 . . . 4 𝑅 ∈ V → (𝑆s (Base‘(1𝑜 mPoly 𝑅))) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))))
1713, 16eqtr4d 2807 . . 3 𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1𝑜 mPoly 𝑅))))
1810, 17pm2.61i 176 . 2 (Poly1𝑅) = (𝑆s (Base‘(1𝑜 mPoly 𝑅)))
191, 18eqtri 2792 1 𝑃 = (𝑆s (Base‘(1𝑜 mPoly 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061  cfv 6031  (class class class)co 6792  1𝑜c1o 7705  Basecbs 16063  s cress 16064   mPoly cmpl 19567  PwSer1cps1 19759  Poly1cpl1 19761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-slot 16067  df-base 16069  df-ress 16071  df-ply1 19766
This theorem is referenced by:  ply1bas  19779  ply1crng  19782  ply1assa  19783  ply1bascl  19787  ply1plusg  19809  ply1vsca  19810  ply1mulr  19811  ply1ring  19832  ply1lmod  19836  ply1sca  19837
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