MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ply1val Structured version   Visualization version   GIF version

Theorem ply1val 22211
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 6907 . . . . . 6 (𝑟 = 𝑅 → (PwSer1𝑟) = (PwSer1𝑅))
3 ply1val.2 . . . . . 6 𝑆 = (PwSer1𝑅)
42, 3eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (PwSer1𝑟) = 𝑆)
5 oveq2 7439 . . . . . 6 (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
65fveq2d 6911 . . . . 5 (𝑟 = 𝑅 → (Base‘(1o mPoly 𝑟)) = (Base‘(1o mPoly 𝑅)))
74, 6oveq12d 7449 . . . 4 (𝑟 = 𝑅 → ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))) = (𝑆s (Base‘(1o mPoly 𝑅))))
8 df-ply1 22199 . . . 4 Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1o mPoly 𝑟))))
9 ovex 7464 . . . 4 (𝑆s (Base‘(1o mPoly 𝑅))) ∈ V
107, 8, 9fvmpt 7016 . . 3 (𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
11 fvprc 6899 . . . . 5 𝑅 ∈ V → (Poly1𝑅) = ∅)
12 ress0 17289 . . . . 5 (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
1311, 12eqtr4di 2793 . . . 4 𝑅 ∈ V → (Poly1𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14 fvprc 6899 . . . . . 6 𝑅 ∈ V → (PwSer1𝑅) = ∅)
153, 14eqtrid 2787 . . . . 5 𝑅 ∈ V → 𝑆 = ∅)
1615oveq1d 7446 . . . 4 𝑅 ∈ V → (𝑆s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
1713, 16eqtr4d 2778 . . 3 𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
1810, 17pm2.61i 182 . 2 (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅)))
191, 18eqtri 2763 1 𝑃 = (𝑆s (Base‘(1o mPoly 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cfv 6563  (class class class)co 7431  1oc1o 8498  Basecbs 17245  s cress 17274   mPoly cmpl 21944  PwSer1cps1 22192  Poly1cpl1 22194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-ply1 22199
This theorem is referenced by:  ply1bas  22212  ply1basOLD  22213  ply1crng  22216  ply1assa  22217  ply1bascl  22221  ply1plusg  22241  ply1vsca  22242  ply1mulr  22243  ply1ring  22265  ply1lmod  22269  ply1sca  22270
  Copyright terms: Public domain W3C validator