Theorem fnpsr 14944

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 14940	. 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 6902	. . . . 5 ⊢ ↑𝑚 Fn (V × V)
3		nn0ex 9522	. . . . 5 ⊢ ℕ0 ∈ V
4		vex 2818	. . . . 5 ⊢ 𝑖 ∈ V
5		fnovex 6091	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℕ0 ∈ V ∧ 𝑖 ∈ V) → (ℕ0 ↑𝑚 𝑖) ∈ V)
6	2, 3, 4, 5	mp3an 1374	. . . 4 ⊢ (ℕ0 ↑𝑚 𝑖) ∈ V
7	6	rabex 4261	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
8		basfn 13358	. . . . . 6 ⊢ Base Fn V
9		vex 2818	. . . . . 6 ⊢ 𝑟 ∈ V
10		funfvex 5692	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
11	10	funfni 5463	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
12	8, 9, 11	mp2an 426	. . . . 5 ⊢ (Base‘𝑟) ∈ V
13		vex 2818	. . . . 5 ⊢ 𝑑 ∈ V
14		fnovex 6091	. . . . 5 ⊢ (( ↑𝑚 Fn (V × V) ∧ (Base‘𝑟) ∈ V ∧ 𝑑 ∈ V) → ((Base‘𝑟) ↑𝑚 𝑑) ∈ V)
15	2, 12, 13, 14	mp3an 1374	. . . 4 ⊢ ((Base‘𝑟) ↑𝑚 𝑑) ∈ V
16		basendxnn 13355	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
17		vex 2818	. . . . . . 7 ⊢ 𝑏 ∈ V
18		opexg 4349	. . . . . . 7 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝑏 ∈ V) → ⟨(Base‘ndx), 𝑏⟩ ∈ V)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ ⟨(Base‘ndx), 𝑏⟩ ∈ V
20		plusgndxnn 13411	. . . . . . 7 ⊢ (+g‘ndx) ∈ ℕ
21	17	a1i 9	. . . . . . . . 9 ⊢ (⊤ → 𝑏 ∈ V)
22	21, 21	ofmresex 6343	. . . . . . . 8 ⊢ (⊤ → ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V)
23	22	mptru 1407	. . . . . . 7 ⊢ ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) ∈ V
24		opexg 4349	. . . . . . 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 13432	. . . . . . . . 9 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
27	26	simpri 113	. . . . . . . 8 ⊢ (.r‘ndx) ∈ ℕ
28	27	elexi 2828	. . . . . . 7 ⊢ (.r‘ndx) ∈ V
29	17, 17	mpoex 6423	. . . . . . 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 4350	. . . . . 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 4570	. . . . . 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 1374	. . . . 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 13453	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
34	33	simpri 113	. . . . . . . 8 ⊢ (Scalar‘ndx) ∈ ℕ
35	34	elexi 2828	. . . . . . 7 ⊢ (Scalar‘ndx) ∈ V
36	35, 9	opex 4350	. . . . . 6 ⊢ ⟨(Scalar‘ndx), 𝑟⟩ ∈ V
37		vscaslid 13463	. . . . . . . . 9 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
38	37	simpri 113	. . . . . . . 8 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
39	38	elexi 2828	. . . . . . 7 ⊢ ( ·𝑠 ‘ndx) ∈ V
40	12, 17	mpoex 6423	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓)) ∈ V
41	39, 40	opex 4350	. . . . . 6 ⊢ ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩ ∈ V
42		tsetndxnn 13489	. . . . . . . 8 ⊢ (TopSet‘ndx) ∈ ℕ
43	42	elexi 2828	. . . . . . 7 ⊢ (TopSet‘ndx) ∈ V
44		topnfn 13544	. . . . . . . . . . 11 ⊢ TopOpen Fn V
45		funfvex 5692	. . . . . . . . . . . 12 ⊢ ((Fun TopOpen ∧ 𝑟 ∈ dom TopOpen) → (TopOpen‘𝑟) ∈ V)
46	45	funfni 5463	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑟 ∈ V) → (TopOpen‘𝑟) ∈ V)
47	44, 9, 46	mp2an 426	. . . . . . . . . 10 ⊢ (TopOpen‘𝑟) ∈ V
48	47	snex 4303	. . . . . . . . 9 ⊢ {(TopOpen‘𝑟)} ∈ V
49	13, 48	xpex 4871	. . . . . . . 8 ⊢ (𝑑 × {(TopOpen‘𝑟)}) ∈ V
50		ptex 13564	. . . . . . . 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 4350	. . . . . 6 ⊢ ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ ∈ V
53		tpexg 4570	. . . . . 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 1374	. . . . 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 4567	. . . 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 4244	. . 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 4244	. 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 6412	1 ⊢ mPwSer Fn (V × V)

Syntax hints:   = wceq 1398  ⊤wtru 1399   ∈ wcel 2205  {crab 2526  Vcvv 2815  ⦋csb 3141   ∪ cun 3212  {csn 3694  {ctp 3696  ⟨cop 3697   class class class wbr 4114   ↦ cmpt 4176   × cxp 4752  ^◡ccnv 4753   ↾ cres 4756   “ cima 4757   Fn wfn 5352  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060   ∘_𝑓 cof 6273   ∘_𝑟 cofr 6274   ↑_𝑚 cmap 6895  Fincfn 6988   ≤ cle 8325   − cmin 8461  ℕcn 9257  ℕ₀cn0 9516  ndxcnx 13296  Slot cslot 13298  Basecbs 13299  +_gcplusg 13377  ._rcmulr 13378  Scalarcsca 13380   ·_𝑠 cvsca 13381  TopSetcts 13383  TopOpenctopn 13540  ∏_tcpt 13555   Σ_g cgsu 13557   mPwSer cmps 14938

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-i2m1 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-map 6897  df-ixp 6947  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-ndx 13302  df-slot 13303  df-base 13305  df-plusg 13390  df-mulr 13391  df-sca 13393  df-vsca 13394  df-tset 13396  df-rest 13541  df-topn 13542  df-topgen 13560  df-pt 13561  df-psr 14940

This theorem is referenced by:  psrelbas  14959  psrplusgg  14962  psradd  14963  psraddcl  14964  mplvalcoe  14974  mplbascoe  14975  fnmpl  14977  mplplusgg  14987