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Theorem imastset 16791
Description: The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasbas.u (𝜑𝑈 = (𝐹s 𝑅))
imasbas.v (𝜑𝑉 = (Base‘𝑅))
imasbas.f (𝜑𝐹:𝑉onto𝐵)
imasbas.r (𝜑𝑅𝑍)
imastset.j 𝐽 = (TopOpen‘𝑅)
imastset.o 𝑂 = (TopSet‘𝑈)
Assertion
Ref Expression
imastset (𝜑𝑂 = (𝐽 qTop 𝐹))

Proof of Theorem imastset
Dummy variables 𝑝 𝑞 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . . 4 (𝜑𝑈 = (𝐹s 𝑅))
2 imasbas.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 eqid 2824 . . . 4 (+g𝑅) = (+g𝑅)
4 eqid 2824 . . . 4 (.r𝑅) = (.r𝑅)
5 eqid 2824 . . . 4 (Scalar‘𝑅) = (Scalar‘𝑅)
6 eqid 2824 . . . 4 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7 eqid 2824 . . . 4 ( ·𝑠𝑅) = ( ·𝑠𝑅)
8 eqid 2824 . . . 4 (·𝑖𝑅) = (·𝑖𝑅)
9 imastset.j . . . 4 𝐽 = (TopOpen‘𝑅)
10 eqid 2824 . . . 4 (dist‘𝑅) = (dist‘𝑅)
11 eqid 2824 . . . 4 (le‘𝑅) = (le‘𝑅)
12 imasbas.f . . . . 5 (𝜑𝐹:𝑉onto𝐵)
13 imasbas.r . . . . 5 (𝜑𝑅𝑍)
14 eqid 2824 . . . . 5 (+g𝑈) = (+g𝑈)
151, 2, 12, 13, 3, 14imasplusg 16786 . . . 4 (𝜑 → (+g𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩})
16 eqid 2824 . . . . 5 (.r𝑈) = (.r𝑈)
171, 2, 12, 13, 4, 16imasmulr 16787 . . . 4 (𝜑 → (.r𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩})
18 eqid 2824 . . . . 5 ( ·𝑠𝑈) = ( ·𝑠𝑈)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 16789 . . . 4 (𝜑 → ( ·𝑠𝑈) = 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))))
20 eqidd 2825 . . . 4 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩})
21 eqidd 2825 . . . 4 (𝜑 → (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹))
22 eqid 2824 . . . . 5 (dist‘𝑈) = (dist‘𝑈)
231, 2, 12, 13, 10, 22imasds 16782 . . . 4 (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 2825 . . . 4 (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 16780 . . 3 (𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
2625fveq2d 6662 . 2 (𝜑 → (TopSet‘𝑈) = (TopSet‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})))
27 imastset.o . 2 𝑂 = (TopSet‘𝑈)
28 ovex 7178 . . 3 (𝐽 qTop 𝐹) ∈ V
29 eqid 2824 . . . . 5 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 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(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
3029imasvalstr 16721 . . . 4 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) Struct ⟨1, 12⟩
31 tsetid 16656 . . . 4 TopSet = Slot (TopSet‘ndx)
32 snsstp1 4733 . . . . 5 {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩} ⊆ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}
33 ssun2 4134 . . . . 5 {⟨(TopSet‘ndx), (𝐽 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(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
3432, 33sstri 3961 . . . 4 {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
3530, 31, 34strfv 16527 . . 3 ((𝐽 qTop 𝐹) ∈ V → (𝐽 qTop 𝐹) = (TopSet‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})))
3628, 35ax-mp 5 . 2 (𝐽 qTop 𝐹) = (TopSet‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ( ·𝑠𝑈)⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
3726, 27, 363eqtr4g 2884 1 (𝜑𝑂 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  cun 3917  {csn 4549  {ctp 4553  cop 4555   ciun 4905  ccnv 5541  ccom 5546  ontowfo 6341  cfv 6343  (class class class)co 7145  1c1 10530  2c2 11685  cdc 12091  ndxcnx 16476  Basecbs 16479  +gcplusg 16561  .rcmulr 16562  Scalarcsca 16564   ·𝑠 cvsca 16565  ·𝑖cip 16566  TopSetcts 16567  lecple 16568  distcds 16570  TopOpenctopn 16691   qTop cqtop 16772  s cimas 16773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-sup 8897  df-inf 8898  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-3 11694  df-4 11695  df-5 11696  df-6 11697  df-7 11698  df-8 11699  df-9 11700  df-n0 11891  df-z 11975  df-dec 12092  df-uz 12237  df-fz 12891  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-plusg 16574  df-mulr 16575  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-imas 16777
This theorem is referenced by:  imasle  16792  imastopn  22321  circtopn  31129
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