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Theorem ply1val 21700
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 6888 . . . . . 6 (𝑟 = 𝑅 → (PwSer1𝑟) = (PwSer1𝑅))
3 ply1val.2 . . . . . 6 𝑆 = (PwSer1𝑅)
42, 3eqtr4di 2791 . . . . 5 (𝑟 = 𝑅 → (PwSer1𝑟) = 𝑆)
5 oveq2 7412 . . . . . 6 (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
65fveq2d 6892 . . . . 5 (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
74, 6oveq12d 7422 . . . 4 (𝑟 = 𝑅 → ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆s (Base‘(1o mPoly 𝑅))))
8 df-ply1 21688 . . . 4 Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9 ovex 7437 . . . 4 (𝑆s (Base‘(1o mPoly 𝑅))) ∈ V
107, 8, 9fvmpt 6994 . . 3 (𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
11 fvprc 6880 . . . . 5 𝑅 ∈ V → (Poly1𝑅) = ∅)
12 ress0 17184 . . . . 5 (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
1311, 12eqtr4di 2791 . . . 4 𝑅 ∈ V → (Poly1𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14 fvprc 6880 . . . . . 6 𝑅 ∈ V → (PwSer1𝑅) = ∅)
153, 14eqtrid 2785 . . . . 5 𝑅 ∈ V → 𝑆 = ∅)
1615oveq1d 7419 . . . 4 𝑅 ∈ V → (𝑆s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
1713, 16eqtr4d 2776 . . 3 𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
1810, 17pm2.61i 182 . 2 (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅)))
191, 18eqtri 2761 1 𝑃 = (𝑆s (Base‘(1o mPoly 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  c0 4321  cfv 6540  (class class class)co 7404  1oc1o 8454  Basecbs 17140  s cress 17169   mPoly cmpl 21441  PwSer1cps1 21681  Poly1cpl1 21683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-nn 12209  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-ply1 21688
This theorem is referenced by:  ply1bas  21701  ply1crng  21704  ply1assa  21705  ply1bascl  21709  ply1plusg  21729  ply1vsca  21730  ply1mulr  21731  ply1ring  21752  ply1lmod  21756  ply1sca  21757
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