Theorem fnpsr 14596

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 14592	. 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 6772	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
3		nn0ex 9343	. . . . 5 ⊢ ℕ0 ∈ V
4		vex 2782	. . . . 5 ⊢ 𝑖 ∈ V
5		fnovex 6007	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝑖 ∈ V) → (ℕ0 ↑𝑚 𝑖) ∈ V)
6	2, 3, 4, 5	mp3an 1352	. . . 4 ⊢ (ℕ0 ↑𝑚 𝑖) ∈ V
7	6	rabex 4207	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
8		basfn 13057	. . . . . 6 ⊢ Base Fn V
9		vex 2782	. . . . . 6 ⊢ 𝑟 ∈ V
10		funfvex 5620	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
11	10	funfni 5399	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
12	8, 9, 11	mp2an 426	. . . . 5 ⊢ (Base‘𝑟) ∈ V
13		vex 2782	. . . . 5 ⊢ 𝑑 ∈ V
14		fnovex 6007	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ (Base‘𝑟) ∈ V ∧ 𝑑 ∈ V) → ((Base‘𝑟) ↑𝑚 𝑑) ∈ V)
15	2, 12, 13, 14	mp3an 1352	. . . 4 ⊢ ((Base‘𝑟) ↑𝑚 𝑑) ∈ V
16		basendxnn 13054	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
17		vex 2782	. . . . . . 7 ⊢ 𝑏 ∈ V
18		opexg 4293	. . . . . . 7 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝑏 ∈ V) → ⟨(Base‘ndx), 𝑏⟩ ∈ V)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ ⟨(Base‘ndx), 𝑏⟩ ∈ V
20		plusgndxnn 13110	. . . . . . 7 ⊢ (+g‘ndx) ∈ ℕ
21	17	a1i 9	. . . . . . . . 9 ⊢ (⊤ → 𝑏 ∈ V)
22	21, 21	ofmresex 6252	. . . . . . . 8 ⊢ (⊤ → ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V)
23	22	mptru 1384	. . . . . . 7 ⊢ ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V
24		opexg 4293	. . . . . . 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 13131	. . . . . . . . 9 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
27	26	simpri 113	. . . . . . . 8 ⊢ (.r‘ndx) ∈ ℕ
28	27	elexi 2792	. . . . . . 7 ⊢ (.r‘ndx) ∈ V
29	17, 17	mpoex 6330	. . . . . . 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 4294	. . . . . 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 4512	. . . . . 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 1352	. . . . 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 13152	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
34	33	simpri 113	. . . . . . . 8 ⊢ (Scalar‘ndx) ∈ ℕ
35	34	elexi 2792	. . . . . . 7 ⊢ (Scalar‘ndx) ∈ V
36	35, 9	opex 4294	. . . . . 6 ⊢ ⟨(Scalar‘ndx), 𝑟⟩ ∈ V
37		vscaslid 13162	. . . . . . . . 9 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
38	37	simpri 113	. . . . . . . 8 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
39	38	elexi 2792	. . . . . . 7 ⊢ ( ·𝑠 ‘ndx) ∈ V
40	12, 17	mpoex 6330	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓)) ∈ V
41	39, 40	opex 4294	. . . . . 6 ⊢ ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩ ∈ V
42		tsetndxnn 13188	. . . . . . . 8 ⊢ (TopSet‘ndx) ∈ ℕ
43	42	elexi 2792	. . . . . . 7 ⊢ (TopSet‘ndx) ∈ V
44		topnfn 13243	. . . . . . . . . . 11 ⊢ TopOpen Fn V
45		funfvex 5620	. . . . . . . . . . . 12 ⊢ ((Fun TopOpen ∧ 𝑟 ∈ dom TopOpen) → (TopOpen‘𝑟) ∈ V)
46	45	funfni 5399	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑟 ∈ V) → (TopOpen‘𝑟) ∈ V)
47	44, 9, 46	mp2an 426	. . . . . . . . . 10 ⊢ (TopOpen‘𝑟) ∈ V
48	47	snex 4248	. . . . . . . . 9 ⊢ {(TopOpen‘𝑟)} ∈ V
49	13, 48	xpex 4811	. . . . . . . 8 ⊢ (𝑑 × {(TopOpen‘𝑟)}) ∈ V
50		ptex 13263	. . . . . . . 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 4294	. . . . . 6 ⊢ ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ ∈ V
53		tpexg 4512	. . . . . 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 1352	. . . . 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 4509	. . . 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 4192	. . 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 4192	. 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 6319	1 ⊢ mPwSer Fn (V × V)

Syntax hints:   = wceq 1375  ⊤wtru 1376   ∈ wcel 2180  {crab 2492  Vcvv 2779  ⦋csb 3104   ∪ cun 3175  {csn 3646  {ctp 3648  ⟨cop 3649   class class class wbr 4062   ↦ cmpt 4124   × cxp 4694  ^◡ccnv 4695   ↾ cres 4698   “ cima 4699   Fn wfn 5289  ‘cfv 5294  (class class class)co 5974   ∈ cmpo 5976   ∘_𝑓 cof 6186   ∘_𝑟 cofr 6187   ↑_𝑚 cmap 6765  Fincfn 6857   ≤ cle 8150   − cmin 8285  ℕcn 9078  ℕ₀cn0 9337  ndxcnx 12995  Slot cslot 12997  Basecbs 12998  +_gcplusg 13076  ._rcmulr 13077  Scalarcsca 13079   ·_𝑠 cvsca 13080  TopSetcts 13082  TopOpenctopn 13239  ∏_tcpt 13254   Σ_g cgsu 13256   mPwSer cmps 14590

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-i2m1 8072

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-of 6188  df-1st 6256  df-2nd 6257  df-map 6767  df-ixp 6816  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-mulr 13090  df-sca 13092  df-vsca 13093  df-tset 13095  df-rest 13240  df-topn 13241  df-topgen 13259  df-pt 13260  df-psr 14592

This theorem is referenced by:  psrelbas  14604  psrplusgg  14607  psradd  14608  psraddcl  14609  mplvalcoe  14619  mplbascoe  14620  fnmpl  14622  mplplusgg  14632