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Theorem imasle 13635
Description: The ordering 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 )
imasle.n  |-  N  =  ( le `  R
)
imasle.l  |-  .<_  =  ( le `  U )
Assertion
Ref Expression
imasle  |-  ( ph  -> 
.<_  =  ( ( F  o.  N )  o.  `' F ) )

Proof of Theorem imasle
Dummy variables  p  q  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasle.l . 2  |-  .<_  =  ( le `  U )
2 imasbas.u . . . . 5  |-  ( ph  ->  U  =  ( F 
"s  R ) )
3 imasbas.v . . . . 5  |-  ( ph  ->  V  =  ( Base `  R ) )
4 eqid 2366 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
5 eqid 2366 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2366 . . . . 5  |-  (Scalar `  R )  =  (Scalar `  R )
7 eqid 2366 . . . . 5  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
8 eqid 2366 . . . . 5  |-  ( .s
`  R )  =  ( .s `  R
)
9 eqid 2366 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
10 eqid 2366 . . . . 5  |-  ( dist `  R )  =  (
dist `  R )
11 imasle.n . . . . 5  |-  N  =  ( le `  R
)
12 imasbas.f . . . . . 6  |-  ( ph  ->  F : V -onto-> B
)
13 imasbas.r . . . . . 6  |-  ( ph  ->  R  e.  Z )
14 eqid 2366 . . . . . 6  |-  ( +g  `  U )  =  ( +g  `  U )
152, 3, 12, 13, 4, 14imasplusg 13630 . . . . 5  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
16 eqid 2366 . . . . . 6  |-  ( .r
`  U )  =  ( .r `  U
)
172, 3, 12, 13, 5, 16imasmulr 13631 . . . . 5  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
18 eqid 2366 . . . . . 6  |-  ( .s
`  U )  =  ( .s `  U
)
192, 3, 12, 13, 6, 7, 8, 18imasvsca 13633 . . . . 5  |-  ( ph  ->  ( .s `  U
)  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
20 eqid 2366 . . . . . 6  |-  (TopSet `  U )  =  (TopSet `  U )
212, 3, 12, 13, 9, 20imastset 13634 . . . . 5  |-  ( ph  ->  (TopSet `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
22 eqid 2366 . . . . . 6  |-  ( dist `  U )  =  (
dist `  U )
232, 3, 12, 13, 10, 22imasds 13626 . . . . 5  |-  ( 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 2367 . . . . 5  |-  ( ph  ->  ( ( F  o.  N )  o.  `' F )  =  ( ( F  o.  N
)  o.  `' F
) )
252, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 21, 23, 24, 12, 13imasval 13624 . . . 4  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } ) )
2625fveq2d 5636 . . 3  |-  ( ph  ->  ( le `  U
)  =  ( le
`  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } ) ) )
27 fof 5557 . . . . . . . 8  |-  ( F : V -onto-> B  ->  F : V --> B )
2812, 27syl 15 . . . . . . 7  |-  ( ph  ->  F : V --> B )
29 fvex 5646 . . . . . . . 8  |-  ( Base `  R )  e.  _V
303, 29syl6eqel 2454 . . . . . . 7  |-  ( ph  ->  V  e.  _V )
31 fex 5869 . . . . . . 7  |-  ( ( F : V --> B  /\  V  e.  _V )  ->  F  e.  _V )
3228, 30, 31syl2anc 642 . . . . . 6  |-  ( ph  ->  F  e.  _V )
33 fvex 5646 . . . . . . 7  |-  ( le
`  R )  e. 
_V
3411, 33eqeltri 2436 . . . . . 6  |-  N  e. 
_V
35 coexg 5318 . . . . . 6  |-  ( ( F  e.  _V  /\  N  e.  _V )  ->  ( F  o.  N
)  e.  _V )
3632, 34, 35sylancl 643 . . . . 5  |-  ( ph  ->  ( F  o.  N
)  e.  _V )
37 cnvexg 5311 . . . . . 6  |-  ( F  e.  _V  ->  `' F  e.  _V )
3832, 37syl 15 . . . . 5  |-  ( ph  ->  `' F  e.  _V )
39 coexg 5318 . . . . 5  |-  ( ( ( F  o.  N
)  e.  _V  /\  `' F  e.  _V )  ->  ( ( F  o.  N )  o.  `' F )  e.  _V )
4036, 38, 39syl2anc 642 . . . 4  |-  ( ph  ->  ( ( F  o.  N )  o.  `' F )  e.  _V )
41 eqid 2366 . . . . . 6  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  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 ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } )
4241imasvalstr 13562 . . . . 5  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } ) Struct  <. 1 , ; 1 2 >.
43 pleid 13509 . . . . 5  |-  le  = Slot  ( le `  ndx )
44 snsstp2 3865 . . . . . 6  |-  { <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. }  C_  {
<. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. }
45 ssun2 3427 . . . . . 6  |-  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  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 ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } )
4644, 45sstri 3274 . . . . 5  |-  { <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. }  C_  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  U
) >. ,  <. ( .r `  ndx ) ,  ( .r `  U
) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } )
4742, 43, 46strfv 13388 . . . 4  |-  ( ( ( F  o.  N
)  o.  `' F
)  e.  _V  ->  ( ( F  o.  N
)  o.  `' F
)  =  ( le
`  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } ) ) )
4840, 47syl 15 . . 3  |-  ( ph  ->  ( ( F  o.  N )  o.  `' F )  =  ( le `  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  U ) >. ,  <. ( .r `  ndx ) ,  ( .r `  U ) >. }  u.  {
<. (Scalar `  ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  ( .s
`  U ) >. } )  u.  { <. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. } ) ) )
4926, 48eqtr4d 2401 . 2  |-  ( ph  ->  ( le `  U
)  =  ( ( F  o.  N )  o.  `' F ) )
501, 49syl5eq 2410 1  |-  ( ph  -> 
.<_  =  ( ( F  o.  N )  o.  `' F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   _Vcvv 2873    u. cun 3236   {csn 3729   {cpr 3730   {ctp 3731   <.cop 3732   `'ccnv 4791    o. ccom 4796   -->wf 5354   -onto->wfo 5356   ` cfv 5358  (class class class)co 5981   1c1 8885   2c2 9942  ;cdc 10275   ndxcnx 13353   Basecbs 13356   +g cplusg 13416   .rcmulr 13417  Scalarcsca 13419   .scvsca 13420  TopSetcts 13422   lecple 13423   distcds 13425   TopOpenctopn 13536    "s cimas 13617
This theorem is referenced by:  imasless  13652  imasleval  13653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-imas 13621
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