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Theorem ply1val 21471
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 6825 . . . . . 6 (𝑟 = 𝑅 → (PwSer1𝑟) = (PwSer1𝑅))
3 ply1val.2 . . . . . 6 𝑆 = (PwSer1𝑅)
42, 3eqtr4di 2794 . . . . 5 (𝑟 = 𝑅 → (PwSer1𝑟) = 𝑆)
5 oveq2 7345 . . . . . 6 (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
65fveq2d 6829 . . . . 5 (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
74, 6oveq12d 7355 . . . 4 (𝑟 = 𝑅 → ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆s (Base‘(1o mPoly 𝑅))))
8 df-ply1 21459 . . . 4 Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9 ovex 7370 . . . 4 (𝑆s (Base‘(1o mPoly 𝑅))) ∈ V
107, 8, 9fvmpt 6931 . . 3 (𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
11 fvprc 6817 . . . . 5 𝑅 ∈ V → (Poly1𝑅) = ∅)
12 ress0 17050 . . . . 5 (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
1311, 12eqtr4di 2794 . . . 4 𝑅 ∈ V → (Poly1𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14 fvprc 6817 . . . . . 6 𝑅 ∈ V → (PwSer1𝑅) = ∅)
153, 14eqtrid 2788 . . . . 5 𝑅 ∈ V → 𝑆 = ∅)
1615oveq1d 7352 . . . 4 𝑅 ∈ V → (𝑆s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
1713, 16eqtr4d 2779 . . 3 𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
1810, 17pm2.61i 182 . 2 (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅)))
191, 18eqtri 2764 1 𝑃 = (𝑆s (Base‘(1o mPoly 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2105  Vcvv 3441  c0 4269  cfv 6479  (class class class)co 7337  1oc1o 8360  Basecbs 17009  s cress 17038   mPoly cmpl 21215  PwSer1cps1 21452  Poly1cpl1 21454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-1cn 11030  ax-addcl 11032
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-nn 12075  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-ply1 21459
This theorem is referenced by:  ply1bas  21472  ply1crng  21475  ply1assa  21476  ply1bascl  21480  ply1plusg  21502  ply1vsca  21503  ply1mulr  21504  ply1ring  21525  ply1lmod  21529  ply1sca  21530
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