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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 = (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 6902 . . . . . 6 (𝑟 = 𝑅 → (PwSer1𝑟) = (PwSer1𝑅))
3 ply1val.2 . . . . . 6 𝑆 = (PwSer1𝑅)
42, 3eqtr4di 2786 . . . . 5 (𝑟 = 𝑅 → (PwSer1𝑟) = 𝑆)
5 oveq2 7434 . . . . . 6 (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
65fveq2d 6906 . . . . 5 (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
74, 6oveq12d 7444 . . . 4 (𝑟 = 𝑅 → ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆s (Base‘(1o mPoly 𝑅))))
8 df-ply1 22108 . . . 4 Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9 ovex 7459 . . . 4 (𝑆s (Base‘(1o mPoly 𝑅))) ∈ V
107, 8, 9fvmpt 7010 . . 3 (𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
11 fvprc 6894 . . . . 5 𝑅 ∈ V → (Poly1𝑅) = ∅)
12 ress0 17231 . . . . 5 (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
1311, 12eqtr4di 2786 . . . 4 𝑅 ∈ V → (Poly1𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14 fvprc 6894 . . . . . 6 𝑅 ∈ V → (PwSer1𝑅) = ∅)
153, 14eqtrid 2780 . . . . 5 𝑅 ∈ V → 𝑆 = ∅)
1615oveq1d 7441 . . . 4 𝑅 ∈ V → (𝑆s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
1713, 16eqtr4d 2771 . . 3 𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
1810, 17pm2.61i 182 . 2 (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅)))
191, 18eqtri 2756 1 𝑃 = (𝑆s (Base‘(1o mPoly 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3473  c0 4326  cfv 6553  (class class class)co 7426  1oc1o 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
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