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Theorem imastset 15379
Description: The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasbas.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasbas.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasbas.f  |-  ( ph  ->  F : V -onto-> B
)
imasbas.r  |-  ( ph  ->  R  e.  Z )
imastset.j  |-  J  =  ( TopOpen `  R )
imastset.o  |-  O  =  (TopSet `  U )
Assertion
Ref Expression
imastset  |-  ( ph  ->  O  =  ( 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  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2429 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2429 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2429 . . . 4  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2429 . . . 4  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2429 . . . 4  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2429 . . . 4  |-  ( .i
`  R )  =  ( .i `  R
)
9 imastset.j . . . 4  |-  J  =  ( TopOpen `  R )
10 eqid 2429 . . . 4  |-  ( dist `  R )  =  (
dist `  R )
11 eqid 2429 . . . 4  |-  ( le
`  R )  =  ( le `  R
)
12 imasbas.f . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
13 imasbas.r . . . . 5  |-  ( ph  ->  R  e.  Z )
14 eqid 2429 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
151, 2, 12, 13, 3, 14imasplusg 15374 . . . 4  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
16 eqid 2429 . . . . 5  |-  ( .r
`  U )  =  ( .r `  U
)
171, 2, 12, 13, 4, 16imasmulr 15375 . . . 4  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
18 eqid 2429 . . . . 5  |-  ( .s
`  U )  =  ( .s `  U
)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 15377 . . . 4  |-  ( ph  ->  ( .s `  U
)  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
20 eqidd 2430 . . . 4  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } )
21 eqidd 2430 . . . 4  |-  ( ph  ->  ( J qTop  F )  =  ( J qTop  F
) )
22 eqid 2429 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
231, 2, 12, 13, 10, 22imasds 15370 . . . 4  |-  ( ph  ->  ( dist `  U
)  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ u  e.  NN  ran  ( z  e.  {
w  e.  ( ( V  X.  V )  ^m  ( 1 ... u ) )  |  ( ( F `  ( 1st `  ( w `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( w `  u
) ) )  =  y  /\  A. v  e.  ( 1 ... (
u  -  1 ) ) ( F `  ( 2nd `  ( w `
 v ) ) )  =  ( F `
 ( 1st `  (
w `  ( v  +  1 ) ) ) ) ) } 
|->  ( RR*s  gsumg  ( (
dist `  R )  o.  z ) ) ) ,  RR* ,  `'  <  ) ) )
24 eqidd 2430 . . . 4  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  =  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 20, 21, 23, 24, 12, 13imasval 15368 . . 3  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) )
2625fveq2d 5885 . 2  |-  ( ph  ->  (TopSet `  U )  =  (TopSet `  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) ) )
27 imastset.o . 2  |-  O  =  (TopSet `  U )
28 ovex 6333 . . 3  |-  ( J qTop 
F )  e.  _V
29 eqid 2429 . . . . 5  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
3029imasvalstr 15309 . . . 4  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) Struct  <. 1 , ; 1 2 >.
31 tsetid 15244 . . . 4  |- TopSet  = Slot  (TopSet ` 
ndx )
32 snsstp1 4154 . . . . 5  |-  { <. (TopSet `  ndx ) ,  ( J qTop  F ) >. }  C_  { <. (TopSet ` 
ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le
`  R ) )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. }
33 ssun2 3636 . . . . 5  |-  { <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le
`  R ) )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. }  C_  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
3432, 33sstri 3479 . . . 4  |-  { <. (TopSet `  ndx ) ,  ( J qTop  F ) >. }  C_  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } )
3530, 31, 34strfv 15120 . . 3  |-  ( ( J qTop  F )  e. 
_V  ->  ( J qTop  F
)  =  (TopSet `  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) ) )
3628, 35ax-mp 5 . 2  |-  ( J qTop 
F )  =  (TopSet `  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( J qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( dist `  U
) >. } ) )
3726, 27, 363eqtr4g 2495 1  |-  ( ph  ->  O  =  ( J qTop 
F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087    u. cun 3440   {csn 4002   {ctp 4006   <.cop 4008   U_ciun 4302   `'ccnv 4853    o. ccom 4858   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   1c1 9539   2c2 10659  ;cdc 11051   ndxcnx 15081   Basecbs 15084   +g cplusg 15152   .rcmulr 15153  Scalarcsca 15155   .scvsca 15156   .icip 15157  TopSetcts 15158   lecple 15159   distcds 15161   TopOpenctopn 15279   qTop cqtop 15360    "s cimas 15361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-imas 15365
This theorem is referenced by:  imasle  15380  imastopn  20666  circtopn  28503
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