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	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrsca.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)

Assertion

Ref	Expression
psrsca	⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))

Proof of Theorem psrsca

Dummy variables 𝑓 ℎ 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrsca.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
2		psrvalstr 21936	. . . 4 ⊢ ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ 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) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}
5		ssun2 4179	. . . . 5 ⊢ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩} ⊆ ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
6	4, 5	sstri 3993	. . . 4 ⊢ {⟨(Scalar‘ndx), 𝑅⟩} ⊆ ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
7	2, 3, 6	strfv 17240	. . 3 ⊢ (𝑅 ∈ 𝑊 → 𝑅 = (Scalar‘({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
8	1, 7	syl 17	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
9		psrsca.s	. . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
10		eqid 2737	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2737	. . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅)
12		eqid 2737	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
13		eqid 2737	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
14		eqid 2737	. . . 4 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
15		eqid 2737	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
16		psrsca.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
17	9, 10, 14, 15, 16	psrbas 21953	. . . 4 ⊢ (𝜑 → (Base‘𝑆) = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
18		eqid 2737	. . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆)
19	9, 15, 11, 18	psrplusg 21956	. . . 4 ⊢ (+g‘𝑆) = ( ∘f (+g‘𝑅) ↾ ((Base‘𝑆) × (Base‘𝑆)))
20		eqid 2737	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
21	9, 15, 12, 20, 14	psrmulr 21962	. . . 4 ⊢ (.r‘𝑆) = (𝑓 ∈ (Base‘𝑆), 𝑧 ∈ (Base‘𝑆) ↦ (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑤} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑧‘(𝑤 ∘f − 𝑥)))))))
22		eqid 2737	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓)) = (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))
23		eqidd 2738	. . . 4 ⊢ (𝜑 → (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) = (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})))
24	9, 10, 11, 12, 13, 14, 17, 19, 21, 22, 23, 16, 1	psrval 21935	. . 3 ⊢ (𝜑 → 𝑆 = ({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}))
25	24	fveq2d 6910	. 2 ⊢ (𝜑 → (Scalar‘𝑆) = (Scalar‘({⟨(Base‘ndx), (Base‘𝑆)⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ (Base‘𝑆) ↦ (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
26	8, 25	eqtr4d 2780	1 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))

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  ℕcn 12266  9c9 12328  ℕ₀cn0 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