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Theorem psrsca 21819
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 21779 . . . 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 17267 . . . 4 Scalar = Slot (Scalar‘ndx)
4 snsstp1 4819 . . . . 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 4173 . . . . 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 3991 . . . 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 17144 . . 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 2731 . . . 4 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2731 . . . 4 (+g𝑅) = (+g𝑅)
12 eqid 2731 . . . 4 (.r𝑅) = (.r𝑅)
13 eqid 2731 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
14 eqid 2731 . . . 4 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
15 eqid 2731 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
16 psrsca.i . . . . 5 (𝜑𝐼𝑉)
179, 10, 14, 15, 16psrbas 21807 . . . 4 (𝜑 → (Base‘𝑆) = ((Base‘𝑅) ↑m { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}))
18 eqid 2731 . . . . 5 (+g𝑆) = (+g𝑆)
199, 15, 11, 18psrplusg 21810 . . . 4 (+g𝑆) = ( ∘f (+g𝑅) ↾ ((Base‘𝑆) × (Base‘𝑆)))
20 eqid 2731 . . . . 5 (.r𝑆) = (.r𝑆)
219, 15, 12, 20, 14psrmulr 21814 . . . 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 2731 . . . 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 2732 . . . 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 21778 . . 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 6895 . 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 2774 1 (𝜑𝑅 = (Scalar‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  {crab 3431  cun 3946  {csn 4628  {ctp 4632  cop 4634   × cxp 5674  ccnv 5675  cima 5679  cfv 6543  (class class class)co 7412  cmpo 7414  f cof 7672  m cmap 8826  Fincfn 8945  1c1 11117  cn 12219  9c9 12281  0cn0 12479  ndxcnx 17133  Basecbs 17151  +gcplusg 17204  .rcmulr 17205  Scalarcsca 17207   ·𝑠 cvsca 17208  TopSetcts 17210  TopOpenctopn 17374  tcpt 17391   mPwSer cmps 21767
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 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-tset 17223  df-psr 21772
This theorem is referenced by:  psrlmod  21832  psrassa  21845  mpllsslem  21870  mplsca  21883  opsrsca  21925  opsrscaOLD  21926  opsrassa  21932  ply1lss  22039  opsrlmod  22088
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