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Theorem ply1val 22195
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 6906 . . . . . 6 (𝑟 = 𝑅 → (PwSer1𝑟) = (PwSer1𝑅))
3 ply1val.2 . . . . . 6 𝑆 = (PwSer1𝑅)
42, 3eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (PwSer1𝑟) = 𝑆)
5 oveq2 7439 . . . . . 6 (𝑟 = 𝑅 → (1o mPoly 𝑟) = (1o mPoly 𝑅))
65fveq2d 6910 . . . . 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 22183 . . . 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 6898 . . . . 5 𝑅 ∈ V → (Poly1𝑅) = ∅)
12 ress0 17289 . . . . 5 (∅ ↾s (Base‘(1o mPoly 𝑅))) = ∅
1311, 12eqtr4di 2795 . . . 4 𝑅 ∈ V → (Poly1𝑅) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
14 fvprc 6898 . . . . . 6 𝑅 ∈ V → (PwSer1𝑅) = ∅)
153, 14eqtrid 2789 . . . . 5 𝑅 ∈ V → 𝑆 = ∅)
1615oveq1d 7446 . . . 4 𝑅 ∈ V → (𝑆s (Base‘(1o mPoly 𝑅))) = (∅ ↾s (Base‘(1o mPoly 𝑅))))
1713, 16eqtr4d 2780 . . 3 𝑅 ∈ V → (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅))))
1810, 17pm2.61i 182 . 2 (Poly1𝑅) = (𝑆s (Base‘(1o mPoly 𝑅)))
191, 18eqtri 2765 1 𝑃 = (𝑆s (Base‘(1o mPoly 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cfv 6561  (class class class)co 7431  1oc1o 8499  Basecbs 17247  s cress 17274   mPoly cmpl 21926  PwSer1cps1 22176  Poly1cpl1 22178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-ply1 22183
This theorem is referenced by:  ply1bas  22196  ply1basOLD  22197  ply1crng  22200  ply1assa  22201  ply1bascl  22205  ply1plusg  22225  ply1vsca  22226  ply1mulr  22227  ply1ring  22249  ply1lmod  22253  ply1sca  22254
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