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Theorem imastset 14582
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 2454 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2454 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2454 . . . 4  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2454 . . . 4  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2454 . . . 4  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2454 . . . 4  |-  ( .i
`  R )  =  ( .i `  R
)
9 imastset.j . . . 4  |-  J  =  ( TopOpen `  R )
10 eqid 2454 . . . 4  |-  ( dist `  R )  =  (
dist `  R )
11 eqid 2454 . . . 4  |-  ( le
`  R )  =  ( le `  R
)
12 imasbas.f . . . . 5  |-  ( ph  ->  F : V -onto-> B
)
13 imasbas.r . . . . 5  |-  ( ph  ->  R  e.  Z )
14 eqid 2454 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
151, 2, 12, 13, 3, 14imasplusg 14577 . . . 4  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
16 eqid 2454 . . . . 5  |-  ( .r
`  U )  =  ( .r `  U
)
171, 2, 12, 13, 4, 16imasmulr 14578 . . . 4  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
18 eqid 2454 . . . . 5  |-  ( .s
`  U )  =  ( .s `  U
)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 14580 . . . 4  |-  ( ph  ->  ( .s `  U
)  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
20 eqidd 2455 . . . 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 2455 . . . 4  |-  ( ph  ->  ( J qTop  F )  =  ( J qTop  F
) )
22 eqid 2454 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
231, 2, 12, 13, 10, 22imasds 14573 . . . 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 2455 . . . 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 14571 . . 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 5806 . 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 6228 . . 3  |-  ( J qTop 
F )  e.  _V
29 eqid 2454 . . . . 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 14512 . . . 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 14448 . . . 4  |- TopSet  = Slot  (TopSet ` 
ndx )
32 snsstp1 4135 . . . . 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 3631 . . . . 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 3476 . . . 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 14329 . . 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 2520 1  |-  ( ph  ->  O  =  ( J qTop 
F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3437   {csn 3988   {ctp 3992   <.cop 3994   U_ciun 4282   `'ccnv 4950    o. ccom 4955   -onto->wfo 5527   ` cfv 5529  (class class class)co 6203   1c1 9397   2c2 10485  ;cdc 10869   ndxcnx 14292   Basecbs 14295   +g cplusg 14360   .rcmulr 14361  Scalarcsca 14363   .scvsca 14364   .icip 14365  TopSetcts 14366   lecple 14367   distcds 14369   TopOpenctopn 14482   qTop cqtop 14563    "s cimas 14564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-plusg 14373  df-mulr 14374  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-imas 14568
This theorem is referenced by:  imasle  14583  imastopn  19428
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