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Theorem ply1val 19504
 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 6158 . . . . . 6 (𝑟 = 𝑅 → (PwSer1𝑟) = (PwSer1𝑅))
3 ply1val.2 . . . . . 6 𝑆 = (PwSer1𝑅)
42, 3syl6eqr 2673 . . . . 5 (𝑟 = 𝑅 → (PwSer1𝑟) = 𝑆)
5 oveq2 6623 . . . . . 6 (𝑟 = 𝑅 → (1𝑜 mPoly 𝑟) = (1𝑜 mPoly 𝑅))
65fveq2d 6162 . . . . 5 (𝑟 = 𝑅 → (Base‘(1𝑜 mPoly 𝑟)) = (Base‘(1𝑜 mPoly 𝑅)))
74, 6oveq12d 6633 . . . 4 (𝑟 = 𝑅 → ((PwSer1𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))) = (𝑆s (Base‘(1𝑜 mPoly 𝑅))))
8 df-ply1 19492 . . . 4 Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))))
9 ovex 6643 . . . 4 (𝑆s (Base‘(1𝑜 mPoly 𝑅))) ∈ V
107, 8, 9fvmpt 6249 . . 3 (𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1𝑜 mPoly 𝑅))))
11 fvprc 6152 . . . . 5 𝑅 ∈ V → (Poly1𝑅) = ∅)
12 ress0 15874 . . . . 5 (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))) = ∅
1311, 12syl6eqr 2673 . . . 4 𝑅 ∈ V → (Poly1𝑅) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))))
14 fvprc 6152 . . . . . 6 𝑅 ∈ V → (PwSer1𝑅) = ∅)
153, 14syl5eq 2667 . . . . 5 𝑅 ∈ V → 𝑆 = ∅)
1615oveq1d 6630 . . . 4 𝑅 ∈ V → (𝑆s (Base‘(1𝑜 mPoly 𝑅))) = (∅ ↾s (Base‘(1𝑜 mPoly 𝑅))))
1713, 16eqtr4d 2658 . . 3 𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1𝑜 mPoly 𝑅))))
1810, 17pm2.61i 176 . 2 (Poly1𝑅) = (𝑆s (Base‘(1𝑜 mPoly 𝑅)))
191, 18eqtri 2643 1 𝑃 = (𝑆s (Base‘(1𝑜 mPoly 𝑅)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1480   ∈ wcel 1987  Vcvv 3190  ∅c0 3897  ‘cfv 5857  (class class class)co 6615  1𝑜c1o 7513  Basecbs 15800   ↾s cress 15801   mPoly cmpl 19293  PwSer1cps1 19485  Poly1cpl1 19487 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-slot 15804  df-base 15805  df-ress 15807  df-ply1 19492 This theorem is referenced by:  ply1bas  19505  ply1crng  19508  ply1assa  19509  ply1bascl  19513  ply1plusg  19535  ply1vsca  19536  ply1mulr  19537  ply1ring  19558  ply1lmod  19562  ply1sca  19563
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