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Theorem imasip 16121
 Description: The inner product of an image structure. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
imasbas.u (𝜑𝑈 = (𝐹s 𝑅))
imasbas.v (𝜑𝑉 = (Base‘𝑅))
imasbas.f (𝜑𝐹:𝑉onto𝐵)
imasbas.r (𝜑𝑅𝑍)
imasip.i , = (·𝑖𝑅)
imasip.w 𝐼 = (·𝑖𝑈)
Assertion
Ref Expression
imasip (𝜑𝐼 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (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.
StepHypRef Expression
1 imasbas.u . . 3 (𝜑𝑈 = (𝐹s 𝑅))
2 imasbas.v . . 3 (𝜑𝑉 = (Base‘𝑅))
3 eqid 2621 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2621 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2621 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
6 eqid 2621 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7 eqid 2621 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
8 imasip.i . . 3 , = (·𝑖𝑅)
9 eqid 2621 . . 3 (TopOpen‘𝑅) = (TopOpen‘𝑅)
10 eqid 2621 . . 3 (dist‘𝑅) = (dist‘𝑅)
11 eqid 2621 . . 3 (le‘𝑅) = (le‘𝑅)
12 imasbas.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
13 imasbas.r . . . 4 (𝜑𝑅𝑍)
14 eqid 2621 . . . 4 (+g𝑈) = (+g𝑈)
151, 2, 12, 13, 3, 14imasplusg 16117 . . 3 (𝜑 → (+g𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩})
16 eqid 2621 . . . 4 (.r𝑈) = (.r𝑈)
171, 2, 12, 13, 4, 16imasmulr 16118 . . 3 (𝜑 → (.r𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩})
18 eqid 2621 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 16120 . . 3 (𝜑 → ( ·𝑠𝑈) = 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))))
20 eqidd 2622 . . 3 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩})
21 eqidd 2622 . . 3 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
22 eqid 2621 . . . 4 (dist‘𝑈) = (dist‘𝑈)
231, 2, 12, 13, 10, 22imasds 16113 . . 3 (𝜑 → (dist‘𝑈) = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑𝑚 (1...𝑢)) ∣ ((𝐹‘(1st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑤𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2nd ‘(𝑤𝑣))) = (𝐹‘(1st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑧))), ℝ*, < )))
24 eqidd 2622 . . 3 (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹))
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 20, 21, 23, 24, 12, 13imasval 16111 . 2 (𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
26 eqid 2621 . . 3 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
2726imasvalstr 16052 . 2 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) Struct ⟨1, 12⟩
28 ipid 15963 . 2 ·𝑖 = Slot (·𝑖‘ndx)
29 snsstp3 4324 . . . 4 {⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}
30 ssun2 3761 . . . 4 {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩})
3129, 30sstri 3597 . . 3 {⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩})
32 ssun1 3760 . . 3 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
3331, 32sstri 3597 . 2 {⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
34 fvex 6168 . . . 4 (Base‘𝑅) ∈ V
352, 34syl6eqel 2706 . . 3 (𝜑𝑉 ∈ V)
36 snex 4879 . . . . . 6 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V
3736rgenw 2920 . . . . 5 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V
38 iunexg 7104 . . . . 5 ((𝑉 ∈ V ∧ ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V) → 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
3935, 37, 38sylancl 693 . . . 4 (𝜑 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
4039ralrimivw 2963 . . 3 (𝜑 → ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
41 iunexg 7104 . . 3 ((𝑉 ∈ V ∧ ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V) → 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
4235, 40, 41syl2anc 692 . 2 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩} ∈ V)
43 imasip.w . 2 𝐼 = (·𝑖𝑈)
4425, 27, 28, 33, 42, 43strfv3 15848 1 (𝜑𝐼 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝 , 𝑞)⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ∀wral 2908  Vcvv 3190   ∪ cun 3558  {csn 4155  {ctp 4159  ⟨cop 4161  ∪ ciun 4492  ◡ccnv 5083   ∘ ccom 5088  –onto→wfo 5855  ‘cfv 5857  (class class class)co 6615  1c1 9897  2c2 11030  ;cdc 11453  ndxcnx 15797  Basecbs 15800  +gcplusg 15881  .rcmulr 15882  Scalarcsca 15884   ·𝑠 cvsca 15885  ·𝑖cip 15886  TopSetcts 15887  lecple 15888  distcds 15890  TopOpenctopn 16022   qTop cqtop 16103   “s cimas 16104 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-imas 16108 This theorem is referenced by: (None)
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