Theorem fnpsr 14164

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 14161	. 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 6711	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
3		nn0ex 9249	. . . . 5 ⊢ ℕ0 ∈ V
4		vex 2763	. . . . 5 ⊢ 𝑖 ∈ V
5		fnovex 5952	. . . . 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 4174	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
8		basfn 12679	. . . . . 6 ⊢ Base Fn V
9		vex 2763	. . . . . 6 ⊢ 𝑟 ∈ V
10		funfvex 5572	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
11	10	funfni 5355	. . . . . 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 5952	. . . . 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 12677	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
17		vex 2763	. . . . . . 7 ⊢ 𝑏 ∈ V
18		opexg 4258	. . . . . . 7 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝑏 ∈ V) → ⟨(Base‘ndx), 𝑏⟩ ∈ V)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ ⟨(Base‘ndx), 𝑏⟩ ∈ V
20		plusgndxnn 12732	. . . . . . 7 ⊢ (+g‘ndx) ∈ ℕ
21	17	a1i 9	. . . . . . . . 9 ⊢ (⊤ → 𝑏 ∈ V)
22	21, 21	ofmresex 6191	. . . . . . . 8 ⊢ (⊤ → ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V)
23	22	mptru 1373	. . . . . . 7 ⊢ ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V
24		opexg 4258	. . . . . . 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 12752	. . . . . . . . 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 6269	. . . . . . 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 4259	. . . . . 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 4476	. . . . . 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 12773	. . . . . . . . 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 4259	. . . . . 6 ⊢ ⟨(Scalar‘ndx), 𝑟⟩ ∈ V
37		vscaslid 12783	. . . . . . . . 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 6269	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓)) ∈ V
41	39, 40	opex 4259	. . . . . 6 ⊢ ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩ ∈ V
42		tsetndxnn 12809	. . . . . . . 8 ⊢ (TopSet‘ndx) ∈ ℕ
43	42	elexi 2772	. . . . . . 7 ⊢ (TopSet‘ndx) ∈ V
44		topnfn 12858	. . . . . . . . . . 11 ⊢ TopOpen Fn V
45		funfvex 5572	. . . . . . . . . . . 12 ⊢ ((Fun TopOpen ∧ 𝑟 ∈ dom TopOpen) → (TopOpen‘𝑟) ∈ V)
46	45	funfni 5355	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑟 ∈ V) → (TopOpen‘𝑟) ∈ V)
47	44, 9, 46	mp2an 426	. . . . . . . . . 10 ⊢ (TopOpen‘𝑟) ∈ V
48	47	snex 4215	. . . . . . . . 9 ⊢ {(TopOpen‘𝑟)} ∈ V
49	13, 48	xpex 4775	. . . . . . . 8 ⊢ (𝑑 × {(TopOpen‘𝑟)}) ∈ V
50		ptex 12878	. . . . . . . 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 4259	. . . . . 6 ⊢ ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ ∈ V
53		tpexg 4476	. . . . . 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 4473	. . . 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 4159	. . 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 4159	. 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 6258	1 ⊢ mPwSer Fn (V × V)

Syntax hints:   = wceq 1364  ⊤wtru 1365   ∈ wcel 2164  {crab 2476  Vcvv 2760  ⦋csb 3081   ∪ cun 3152  {csn 3619  {ctp 3621  ⟨cop 3622   class class class wbr 4030   ↦ cmpt 4091   × cxp 4658  ^◡ccnv 4659   ↾ cres 4662   “ cima 4663   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919   ∈ cmpo 5921   ∘_𝑓 cof 6130   ∘_𝑟 cofr 6131   ↑_𝑚 cmap 6704  Fincfn 6796   ≤ cle 8057   − cmin 8192  ℕcn 8984  ℕ₀cn0 9243  ndxcnx 12618  Slot cslot 12620  Basecbs 12621  +_gcplusg 12698  ._rcmulr 12699  Scalarcsca 12701   ·_𝑠 cvsca 12702  TopSetcts 12704  TopOpenctopn 12854  ∏_tcpt 12869   Σ_g cgsu 12871   mPwSer cmps 14160

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 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-i2m1 7979

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-of 6132  df-1st 6195  df-2nd 6196  df-map 6706  df-ixp 6755  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-n0 9244  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-mulr 12712  df-sca 12714  df-vsca 12715  df-tset 12717  df-rest 12855  df-topn 12856  df-topgen 12874  df-pt 12875  df-psr 14161

This theorem is referenced by:  psrelbas  14171  psrplusgg  14173  psradd  14174  psraddcl  14175