Theorem imasip 17576

Description: The inner product of an image structure. (Contributed by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
imasbas.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasbas.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasbas.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasip.i	⊢ , = (·𝑖‘𝑅)
imasip.w	⊢ 𝐼 = (·𝑖‘𝑈)

Assertion

Ref	Expression
imasip	⊢ (𝜑 → 𝐼 = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩})

Distinct variable groups:   𝑞,𝑝,𝐹   𝑅,𝑝,𝑞   𝜑,𝑝,𝑞   𝑉,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑞,𝑝)   𝑈(𝑞,𝑝)   , (𝑞,𝑝)   𝐼(𝑞,𝑝)   𝑍(𝑞,𝑝)

Proof of Theorem imasip

Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasbas.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasbas.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2763	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2763	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2763	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
6		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
8		imasip.i	. . 3 ⊢ , = (·𝑖‘𝑅)
9		eqid 2763	. . 3 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10		eqid 2763	. . 3 ⊢ (dist‘𝑅) = (dist‘𝑅)
11		eqid 2763	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
12		imasbas.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
13		imasbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
14		eqid 2763	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
15	1, 2, 12, 13, 3, 14	imasplusg 17572	. . 3 ⊢ (𝜑 → (+g‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
16		eqid 2763	. . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈)
17	1, 2, 12, 13, 4, 16	imasmulr 17573	. . 3 ⊢ (𝜑 → (.r‘𝑈) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
18		eqid 2763	. . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
19	1, 2, 12, 13, 5, 6, 7, 18	imasvsca 17575	. . 3 ⊢ (𝜑 → ( ·𝑠 ‘𝑈) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞))))
20		eqidd 2764	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩})
21		eqidd 2764	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
22		eqid 2763	. . . 4 ⊢ (dist‘𝑈) = (dist‘𝑈)
23	1, 2, 12, 13, 10, 22	imasds 17568	. . 3 ⊢ (𝜑 → (dist‘𝑈) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑m (1...𝑢)) ∣ ((𝐹‘(1st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑤‘𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2nd ‘(𝑤‘𝑣))) = (𝐹‘(1st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑧))), ℝ*, < )))
24		eqidd 2764	. . 3 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹))
25	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 20, 21, 23, 24, 12, 13	imasval 17566	. 2 ⊢ (𝜑 → 𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
26		eqid 2763	. . 3 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
27	26	imasvalstr 17505	. 2 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) Struct ⟨1, ;12⟩
28		ipid 17385	. 2 ⊢ ·𝑖 = Slot (·𝑖‘ndx)
29		snsstp3 4785	. . . 4 ⊢ {⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}
30		ssun2 4133	. . . 4 ⊢ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩})
31	29, 30	sstri 3947	. . 3 ⊢ {⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩})
32		ssun1 4132	. . 3 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
33	31, 32	sstri 3947	. 2 ⊢ {⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑈)⟩, ⟨(.r‘ndx), (.r‘𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑈)⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
34		fvex 6896	. . . 4 ⊢ (Base‘𝑅) ∈ V
35	2, 34	eqeltrdi 2871	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
36		snex 5412	. . . . . 6 ⊢ {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V
37	36	rgenw 3083	. . . . 5 ⊢ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V
38		iunexg 7961	. . . . 5 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
39	35, 37, 38	sylancl 597	. . . 4 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
40	39	ralrimivw 3161	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
41		iunexg 7961	. . 3 ⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
42	35, 40, 41	syl2anc 595	. 2 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
43		imasip.w	. 2 ⊢ 𝐼 = (·𝑖‘𝑈)
44	25, 27, 28, 33, 42, 43	strfv3 17265	1 ⊢ (𝜑 → 𝐼 = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝 , 𝑞)⟩})

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∪ cun 3904  {csn 4590  {ctp 4594  ⟨cop 4596  ∪ ciun 4957  ^◡ccnv 5662   ∘ ccom 5667  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7412  1c1 11102  2c2 12296  ;cdc 12712  ndxcnx 17254  Basecbs 17270  +_gcplusg 17311  ._rcmulr 17312  Scalarcsca 17314   ·_𝑠 cvsca 17315  ·_𝑖cip 17316  TopSetcts 17317  lecple 17318  distcds 17320  TopOpenctopn 17475   qTop cqtop 17558   “_s cimas 17559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-struct 17208  df-slot 17243  df-ndx 17255  df-base 17271  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-imas 17563

This theorem is referenced by: (None)