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Theorem psrsca 21967
Description: The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
psrsca.s 𝑆 = (𝐼 mPwSer 𝑅)
psrsca.i (𝜑𝐼𝑉)
psrsca.r (𝜑𝑅𝑊)
Assertion
Ref Expression
psrsca (𝜑𝑅 = (Scalar‘𝑆))

Proof of Theorem psrsca
Dummy variables 𝑓 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrsca.r . . 3 (𝜑𝑅𝑊)
2 psrvalstr 21936 . . . 4 ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g𝑆)⟩, ⟨(.r‘ndx), (.r𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}) Struct ⟨1, 9⟩
3 scaid 17359 . . . 4 Scalar = Slot (Scalar‘ndx)
4 snsstp1 4816 . . . . 5 {⟨(Scalar‘ndx), 𝑅⟩} ⊆ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}
5 ssun2 4179 . . . . 5 {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩} ⊆ ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g𝑆)⟩, ⟨(.r‘ndx), (.r𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
64, 5sstri 3993 . . . 4 {⟨(Scalar‘ndx), 𝑅⟩} ⊆ ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g𝑆)⟩, ⟨(.r‘ndx), (.r𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
72, 3, 6strfv 17240 . . 3 (𝑅𝑊𝑅 = (Scalar‘({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g𝑆)⟩, ⟨(.r‘ndx), (.r𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
81, 7syl 17 . 2 (𝜑𝑅 = (Scalar‘({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g𝑆)⟩, ⟨(.r‘ndx), (.r𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
9 psrsca.s . . . 4 𝑆 = (𝐼 mPwSer 𝑅)
10 eqid 2737 . . . 4 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2737 . . . 4 (+g𝑅) = (+g𝑅)
12 eqid 2737 . . . 4 (.r𝑅) = (.r𝑅)
13 eqid 2737 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
14 eqid 2737 . . . 4 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
15 eqid 2737 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
16 psrsca.i . . . . 5 (𝜑𝐼𝑉)
179, 10, 14, 15, 16psrbas 21953 . . . 4 (𝜑 → (Base‘𝑆) = ((Base‘𝑅) ↑m { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}))
18 eqid 2737 . . . . 5 (+g𝑆) = (+g𝑆)
199, 15, 11, 18psrplusg 21956 . . . 4 (+g𝑆) = ( ∘f (+g𝑅) ↾ ((Base‘𝑆) × (Base‘𝑆)))
20 eqid 2737 . . . . 5 (.r𝑆) = (.r𝑆)
219, 15, 12, 20, 14psrmulr 21962 . . . 4 (.r𝑆) = (𝑓 ∈ (Base‘𝑆), 𝑧 ∈ (Base‘𝑆) ↦ (𝑤 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦r𝑤} ↦ ((𝑓𝑥)(.r𝑅)(𝑧‘(𝑤f𝑥)))))))
22 eqid 2737 . . . 4 (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓)) = (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))
23 eqidd 2738 . . . 4 (𝜑 → (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) = (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})))
249, 10, 11, 12, 13, 14, 17, 19, 21, 22, 23, 16, 1psrval 21935 . . 3 (𝜑𝑆 = ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g𝑆)⟩, ⟨(.r‘ndx), (.r𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}))
2524fveq2d 6910 . 2 (𝜑 → (Scalar‘𝑆) = (Scalar‘({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g𝑆)⟩, ⟨(.r‘ndx), (.r𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
268, 25eqtr4d 2780 1 (𝜑𝑅 = (Scalar‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  cun 3949  {csn 4626  {ctp 4630  cop 4632   × cxp 5683  ccnv 5684  cima 5688  cfv 6561  (class class class)co 7431  cmpo 7433  f cof 7695  m cmap 8866  Fincfn 8985  1c1 11156  cn 12266  9c9 12328  0cn0 12526  ndxcnx 17230  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  TopSetcts 17303  TopOpenctopn 17466  tcpt 17483   mPwSer cmps 21924
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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
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-nel 3047  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-tp 4631  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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-psr 21929
This theorem is referenced by:  psrlmod  21980  psrassa  21993  psrascl  21999  psrasclcl  22000  mpllsslem  22020  mplsca  22033  opsrsca  22077  opsrscaOLD  22078  opsrassa  22084  psdascl  22172  ply1lss  22198  opsrlmod  22247
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