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Theorem psrsca 21357
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 21318 . . . 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 17196 . . . 4 Scalar = Slot (Scalar‘ndx)
4 snsstp1 4776 . . . . 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 4133 . . . . 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 3953 . . . 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 17076 . . 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 2736 . . . 4 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2736 . . . 4 (+g𝑅) = (+g𝑅)
12 eqid 2736 . . . 4 (.r𝑅) = (.r𝑅)
13 eqid 2736 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
14 eqid 2736 . . . 4 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
15 eqid 2736 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
16 psrsca.i . . . . 5 (𝜑𝐼𝑉)
179, 10, 14, 15, 16psrbas 21346 . . . 4 (𝜑 → (Base‘𝑆) = ((Base‘𝑅) ↑m { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}))
18 eqid 2736 . . . . 5 (+g𝑆) = (+g𝑆)
199, 15, 11, 18psrplusg 21349 . . . 4 (+g𝑆) = ( ∘f (+g𝑅) ↾ ((Base‘𝑆) × (Base‘𝑆)))
20 eqid 2736 . . . . 5 (.r𝑆) = (.r𝑆)
219, 15, 12, 20, 14psrmulr 21352 . . . 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 2736 . . . 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 2737 . . . 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 21317 . . 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 6846 . 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 2779 1 (𝜑𝑅 = (Scalar‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3407  cun 3908  {csn 4586  {ctp 4590  cop 4592   × cxp 5631  ccnv 5632  cima 5636  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615  m cmap 8765  Fincfn 8883  1c1 11052  cn 12153  9c9 12215  0cn0 12413  ndxcnx 17065  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  TopSetcts 17139  TopOpenctopn 17303  tcpt 17320   mPwSer cmps 21306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-tset 17152  df-psr 21311
This theorem is referenced by:  psrlmod  21370  psrassa  21383  mpllsslem  21406  mplsca  21417  opsrsca  21460  opsrscaOLD  21461  opsrassa  21467  ply1lss  21567  opsrlmod  21617
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