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Theorem psrsca 19210
 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 19184 . . . 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 15837 . . . 4 Scalar = Slot (Scalar‘ndx)
4 snsstp1 4287 . . . . 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 3739 . . . . 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 3577 . . . 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 15735 . . 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 2610 . . . 4 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2610 . . . 4 (+g𝑅) = (+g𝑅)
12 eqid 2610 . . . 4 (.r𝑅) = (.r𝑅)
13 eqid 2610 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
14 eqid 2610 . . . 4 { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
15 eqid 2610 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
16 psrsca.i . . . . 5 (𝜑𝐼𝑉)
179, 10, 14, 15, 16psrbas 19199 . . . 4 (𝜑 → (Base‘𝑆) = ((Base‘𝑅) ↑𝑚 { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}))
18 eqid 2610 . . . . 5 (+g𝑆) = (+g𝑆)
199, 15, 11, 18psrplusg 19202 . . . 4 (+g𝑆) = ( ∘𝑓 (+g𝑅) ↾ ((Base‘𝑆) × (Base‘𝑆)))
20 eqid 2610 . . . . 5 (.r𝑆) = (.r𝑆)
219, 15, 12, 20, 14psrmulr 19205 . . . 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 2610 . . . 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 2611 . . . 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 19183 . . 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 6107 . 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 2647 1 (𝜑𝑅 = (Scalar‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900   ∪ cun 3538  {csn 4125  {ctp 4129  ⟨cop 4131   × cxp 5036  ◡ccnv 5037   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ∘𝑓 cof 6793   ↑𝑚 cmap 7744  Fincfn 7841  1c1 9816  ℕcn 10897  9c9 10954  ℕ0cn0 11169  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  TopSetcts 15774  TopOpenctopn 15905  ∏tcpt 15922   mPwSer cmps 19172 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-psr 19177 This theorem is referenced by:  psrlmod  19222  psrassa  19235  mpllsslem  19256  mplsca  19266  opsrsca  19304  opsrassa  19310  ply1lss  19387  opsrlmod  19437
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