Theorem fnpsr 14153

Description: The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)

Assertion

Ref	Expression
fnpsr	⊢ mPwSer Fn (V × V)

Proof of Theorem fnpsr

Dummy variables 𝑏 𝑑 𝑓 𝑔 ℎ 𝑖 𝑘 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-psr 14150	. 2 ⊢ mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
2		fnmap 6709	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
3		nn0ex 9246	. . . . 5 ⊢ ℕ0 ∈ V
4		vex 2763	. . . . 5 ⊢ 𝑖 ∈ V
5		fnovex 5951	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝑖 ∈ V) → (ℕ0 ↑𝑚 𝑖) ∈ V)
6	2, 3, 4, 5	mp3an 1348	. . . 4 ⊢ (ℕ0 ↑𝑚 𝑖) ∈ V
7	6	rabex 4173	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
8		basfn 12676	. . . . . 6 ⊢ Base Fn V
9		vex 2763	. . . . . 6 ⊢ 𝑟 ∈ V
10		funfvex 5571	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
11	10	funfni 5354	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
12	8, 9, 11	mp2an 426	. . . . 5 ⊢ (Base‘𝑟) ∈ V
13		vex 2763	. . . . 5 ⊢ 𝑑 ∈ V
14		fnovex 5951	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ (Base‘𝑟) ∈ V ∧ 𝑑 ∈ V) → ((Base‘𝑟) ↑𝑚 𝑑) ∈ V)
15	2, 12, 13, 14	mp3an 1348	. . . 4 ⊢ ((Base‘𝑟) ↑𝑚 𝑑) ∈ V
16		basendxnn 12674	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
17		vex 2763	. . . . . . 7 ⊢ 𝑏 ∈ V
18		opexg 4257	. . . . . . 7 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝑏 ∈ V) → ⟨(Base‘ndx), 𝑏⟩ ∈ V)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ ⟨(Base‘ndx), 𝑏⟩ ∈ V
20		plusgndxnn 12729	. . . . . . 7 ⊢ (+g‘ndx) ∈ ℕ
21	17	a1i 9	. . . . . . . . 9 ⊢ (⊤ → 𝑏 ∈ V)
22	21, 21	ofmresex 6189	. . . . . . . 8 ⊢ (⊤ → ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V)
23	22	mptru 1373	. . . . . . 7 ⊢ ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V
24		opexg 4257	. . . . . . 7 ⊢ (((+g‘ndx) ∈ ℕ ∧ ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V) → ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩ ∈ V)
25	20, 23, 24	mp2an 426	. . . . . 6 ⊢ ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩ ∈ V
26		mulrslid 12749	. . . . . . . . 9 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
27	26	simpri 113	. . . . . . . 8 ⊢ (.r‘ndx) ∈ ℕ
28	27	elexi 2772	. . . . . . 7 ⊢ (.r‘ndx) ∈ V
29	17, 17	mpoex 6267	. . . . . . 7 ⊢ (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) ∈ V
30	28, 29	opex 4258	. . . . . 6 ⊢ ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩ ∈ V
31		tpexg 4475	. . . . . 6 ⊢ ((⟨(Base‘ndx), 𝑏⟩ ∈ V ∧ ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩ ∈ V ∧ ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩ ∈ V) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∈ V)
32	19, 25, 30, 31	mp3an 1348	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∈ V
33		scaslid 12770	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
34	33	simpri 113	. . . . . . . 8 ⊢ (Scalar‘ndx) ∈ ℕ
35	34	elexi 2772	. . . . . . 7 ⊢ (Scalar‘ndx) ∈ V
36	35, 9	opex 4258	. . . . . 6 ⊢ ⟨(Scalar‘ndx), 𝑟⟩ ∈ V
37		vscaslid 12780	. . . . . . . . 9 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
38	37	simpri 113	. . . . . . . 8 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
39	38	elexi 2772	. . . . . . 7 ⊢ ( ·𝑠 ‘ndx) ∈ V
40	12, 17	mpoex 6267	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓)) ∈ V
41	39, 40	opex 4258	. . . . . 6 ⊢ ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩ ∈ V
42		tsetndxnn 12806	. . . . . . . 8 ⊢ (TopSet‘ndx) ∈ ℕ
43	42	elexi 2772	. . . . . . 7 ⊢ (TopSet‘ndx) ∈ V
44		topnfn 12855	. . . . . . . . . . 11 ⊢ TopOpen Fn V
45		funfvex 5571	. . . . . . . . . . . 12 ⊢ ((Fun TopOpen ∧ 𝑟 ∈ dom TopOpen) → (TopOpen‘𝑟) ∈ V)
46	45	funfni 5354	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑟 ∈ V) → (TopOpen‘𝑟) ∈ V)
47	44, 9, 46	mp2an 426	. . . . . . . . . 10 ⊢ (TopOpen‘𝑟) ∈ V
48	47	snex 4214	. . . . . . . . 9 ⊢ {(TopOpen‘𝑟)} ∈ V
49	13, 48	xpex 4774	. . . . . . . 8 ⊢ (𝑑 × {(TopOpen‘𝑟)}) ∈ V
50		ptex 12875	. . . . . . . 8 ⊢ ((𝑑 × {(TopOpen‘𝑟)}) ∈ V → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) ∈ V)
51	49, 50	ax-mp 5	. . . . . . 7 ⊢ (∏t‘(𝑑 × {(TopOpen‘𝑟)})) ∈ V
52	43, 51	opex 4258	. . . . . 6 ⊢ ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ ∈ V
53		tpexg 4475	. . . . . 6 ⊢ ((⟨(Scalar‘ndx), 𝑟⟩ ∈ V ∧ ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩ ∈ V ∧ ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ ∈ V) → {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩} ∈ V)
54	36, 41, 52, 53	mp3an 1348	. . . . 5 ⊢ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩} ∈ V
55	32, 54	unex 4472	. . . 4 ⊢ ({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) ∈ V
56	15, 55	csbexa 4158	. . 3 ⊢ ⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) ∈ V
57	7, 56	csbexa 4158	. 2 ⊢ ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) ∈ V
58	1, 57	fnmpoi 6257	1 ⊢ mPwSer Fn (V × V)

Syntax hints:   = wceq 1364  ⊤wtru 1365   ∈ wcel 2164  {crab 2476  Vcvv 2760  ⦋csb 3080   ∪ cun 3151  {csn 3618  {ctp 3620  ⟨cop 3621   class class class wbr 4029   ↦ cmpt 4090   × cxp 4657  ^◡ccnv 4658   ↾ cres 4661   “ cima 4662   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920   ∘_𝑓 cof 6128   ∘_𝑟 cofr 6129   ↑_𝑚 cmap 6702  Fincfn 6794   ≤ cle 8055   − cmin 8190  ℕcn 8982  ℕ₀cn0 9240  ndxcnx 12615  Slot cslot 12617  Basecbs 12618  +_gcplusg 12695  ._rcmulr 12696  Scalarcsca 12698   ·_𝑠 cvsca 12699  TopSetcts 12701  TopOpenctopn 12851  ∏_tcpt 12866   Σ_g cgsu 12868   mPwSer cmps 14149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-i2m1 7977

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130  df-1st 6193  df-2nd 6194  df-map 6704  df-ixp 6753  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-tset 12714  df-rest 12852  df-topn 12853  df-topgen 12871  df-pt 12872  df-psr 14150

This theorem is referenced by:  psrelbas  14160  psrplusgg  14162  psradd  14163  psraddcl  14164