Theorem fnpsr 15053

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 15049	. 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 6929	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
3		nn0ex 9571	. . . . 5 ⊢ ℕ0 ∈ V
4		vex 2824	. . . . 5 ⊢ 𝑖 ∈ V
5		fnovex 6118	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝑖 ∈ V) → (ℕ0 ↑𝑚 𝑖) ∈ V)
6	2, 3, 4, 5	mp3an 1378	. . . 4 ⊢ (ℕ0 ↑𝑚 𝑖) ∈ V
7	6	rabex 4280	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
8		basfn 13413	. . . . . 6 ⊢ Base Fn V
9		vex 2824	. . . . . 6 ⊢ 𝑟 ∈ V
10		funfvex 5712	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
11	10	funfni 5483	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
12	8, 9, 11	mp2an 430	. . . . 5 ⊢ (Base‘𝑟) ∈ V
13		vex 2824	. . . . 5 ⊢ 𝑑 ∈ V
14		fnovex 6118	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ (Base‘𝑟) ∈ V ∧ 𝑑 ∈ V) → ((Base‘𝑟) ↑𝑚 𝑑) ∈ V)
15	2, 12, 13, 14	mp3an 1378	. . . 4 ⊢ ((Base‘𝑟) ↑𝑚 𝑑) ∈ V
16		basendxnn 13410	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
17		vex 2824	. . . . . . 7 ⊢ 𝑏 ∈ V
18		opexg 4368	. . . . . . 7 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝑏 ∈ V) → ⟨(Base‘ndx), 𝑏⟩ ∈ V)
19	16, 17, 18	mp2an 430	. . . . . 6 ⊢ ⟨(Base‘ndx), 𝑏⟩ ∈ V
20		plusgndxnn 13467	. . . . . . 7 ⊢ (+g‘ndx) ∈ ℕ
21	17	a1i 9	. . . . . . . . 9 ⊢ (⊤ → 𝑏 ∈ V)
22	21, 21	ofmresex 6370	. . . . . . . 8 ⊢ (⊤ → ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V)
23	22	mptru 1411	. . . . . . 7 ⊢ ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V
24		opexg 4368	. . . . . . 7 ⊢ (((+g‘ndx) ∈ ℕ ∧ ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V) → ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩ ∈ V)
25	20, 23, 24	mp2an 430	. . . . . 6 ⊢ ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩ ∈ V
26		mulrslid 13488	. . . . . . . . 9 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
27	26	simpri 113	. . . . . . . 8 ⊢ (.r‘ndx) ∈ ℕ
28	27	elexi 2834	. . . . . . 7 ⊢ (.r‘ndx) ∈ V
29	17, 17	mpoex 6450	. . . . . . 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 4369	. . . . . 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 4590	. . . . . 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 1378	. . . . 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 13509	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
34	33	simpri 113	. . . . . . . 8 ⊢ (Scalar‘ndx) ∈ ℕ
35	34	elexi 2834	. . . . . . 7 ⊢ (Scalar‘ndx) ∈ V
36	35, 9	opex 4369	. . . . . 6 ⊢ ⟨(Scalar‘ndx), 𝑟⟩ ∈ V
37		vscaslid 13519	. . . . . . . . 9 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
38	37	simpri 113	. . . . . . . 8 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
39	38	elexi 2834	. . . . . . 7 ⊢ ( ·𝑠 ‘ndx) ∈ V
40	12, 17	mpoex 6450	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓)) ∈ V
41	39, 40	opex 4369	. . . . . 6 ⊢ ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩ ∈ V
42		tsetndxnn 13545	. . . . . . . 8 ⊢ (TopSet‘ndx) ∈ ℕ
43	42	elexi 2834	. . . . . . 7 ⊢ (TopSet‘ndx) ∈ V
44		topnfn 13600	. . . . . . . . . . 11 ⊢ TopOpen Fn V
45		funfvex 5712	. . . . . . . . . . . 12 ⊢ ((Fun TopOpen ∧ 𝑟 ∈ dom TopOpen) → (TopOpen‘𝑟) ∈ V)
46	45	funfni 5483	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑟 ∈ V) → (TopOpen‘𝑟) ∈ V)
47	44, 9, 46	mp2an 430	. . . . . . . . . 10 ⊢ (TopOpen‘𝑟) ∈ V
48	47	snex 4322	. . . . . . . . 9 ⊢ {(TopOpen‘𝑟)} ∈ V
49	13, 48	xpex 4891	. . . . . . . 8 ⊢ (𝑑 × {(TopOpen‘𝑟)}) ∈ V
50		ptex 13620	. . . . . . . 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 4369	. . . . . 6 ⊢ ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ ∈ V
53		tpexg 4590	. . . . . 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 1378	. . . . 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 4587	. . . 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 4262	. . 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 4262	. 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 6439	1 ⊢ mPwSer Fn (V × V)

This proof depends on syntax axioms:   = wceq 1402  ⊤wtru 1403   ∈ wcel 2209  {crab 2532  Vcvv 2821  ⦋csb 3147   ∪ cun 3218  {csn 3709  {ctp 3711  ⟨cop 3712   class class class wbr 4130   ↦ cmpt 4192   × cxp 4772  ^◡ccnv 4773   ↾ cres 4776   “ cima 4777   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087   ∘_𝑓 cof 6300   ∘_𝑟 cofr 6301   ↑_𝑚 cmap 6922  Fincfn 7022   ≤ cle 8361   − cmin 8497  ℕcn 9305  ℕ₀cn0 9565  ndxcnx 13351  Slot cslot 13353  Basecbs 13354  +_gcplusg 13433  ._rcmulr 13434  Scalarcsca 13436   ·_𝑠 cvsca 13437  TopSetcts 13439  TopOpenctopn 13596  ∏_tcpt 13611   Σ_g cgsu 14152   mPwSer cmps 15047

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-i2m1 8284

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-map 6924  df-ixp 6981  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-ndx 13357  df-slot 13358  df-base 13360  df-plusg 13446  df-mulr 13447  df-sca 13449  df-vsca 13450  df-tset 13452  df-rest 13597  df-topn 13598  df-topgen 13616  df-pt 13617  df-psr 15049

This theorem is used by:  psrelbas  15068  psrplusgg  15071  psradd  15072  psraddcl  15073  mplvalcoe  15083  mplbascoe  15084  fnmpl  15086  mplplusgg  15096