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Theorem imasle 13711
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 imasbas.u . . 3  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2412 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2412 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2412 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2412 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2412 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2412 . . 3  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
9 eqid 2412 . . 3  |-  ( dist `  R )  =  (
dist `  R )
10 imasle.n . . 3  |-  N  =  ( le `  R
)
11 imasbas.f . . . 4  |-  ( ph  ->  F : V -onto-> B
)
12 imasbas.r . . . 4  |-  ( ph  ->  R  e.  Z )
13 eqid 2412 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
141, 2, 11, 12, 3, 13imasplusg 13706 . . 3  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
15 eqid 2412 . . . 4  |-  ( .r
`  U )  =  ( .r `  U
)
161, 2, 11, 12, 4, 15imasmulr 13707 . . 3  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
17 eqid 2412 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
181, 2, 11, 12, 5, 6, 7, 17imasvsca 13709 . . 3  |-  ( ph  ->  ( .s `  U
)  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
19 eqid 2412 . . . 4  |-  (TopSet `  U )  =  (TopSet `  U )
201, 2, 11, 12, 8, 19imastset 13710 . . 3  |-  ( ph  ->  (TopSet `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
21 eqid 2412 . . . 4  |-  ( dist `  U )  =  (
dist `  U )
221, 2, 11, 12, 9, 21imasds 13702 . . 3  |-  ( 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* ,  `'  <  ) ) )
23 eqidd 2413 . . 3  |-  ( ph  ->  ( ( F  o.  N )  o.  `' F )  =  ( ( F  o.  N
)  o.  `' F
) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 18, 20, 22, 23, 11, 12imasval 13700 . 2  |-  ( 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 ) >. } ) )
25 eqid 2412 . . 3  |-  ( ( { <. ( 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 ) >. } )
2625imasvalstr 13638 . 2  |-  ( ( { <. ( 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 >.
27 pleid 13585 . 2  |-  le  = Slot  ( le `  ndx )
28 snsstp2 3918 . . 3  |-  { <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. }  C_  {
<. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. }
29 ssun2 3479 . . 3  |-  { <. (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 ) >. } )
3028, 29sstri 3325 . 2  |-  { <. ( 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 ) >. } )
31 fof 5620 . . . . . 6  |-  ( F : V -onto-> B  ->  F : V --> B )
3211, 31syl 16 . . . . 5  |-  ( ph  ->  F : V --> B )
33 fvex 5709 . . . . . 6  |-  ( Base `  R )  e.  _V
342, 33syl6eqel 2500 . . . . 5  |-  ( ph  ->  V  e.  _V )
35 fex 5936 . . . . 5  |-  ( ( F : V --> B  /\  V  e.  _V )  ->  F  e.  _V )
3632, 34, 35syl2anc 643 . . . 4  |-  ( ph  ->  F  e.  _V )
37 fvex 5709 . . . . 5  |-  ( le
`  R )  e. 
_V
3810, 37eqeltri 2482 . . . 4  |-  N  e. 
_V
39 coexg 5379 . . . 4  |-  ( ( F  e.  _V  /\  N  e.  _V )  ->  ( F  o.  N
)  e.  _V )
4036, 38, 39sylancl 644 . . 3  |-  ( ph  ->  ( F  o.  N
)  e.  _V )
41 cnvexg 5372 . . . 4  |-  ( F  e.  _V  ->  `' F  e.  _V )
4236, 41syl 16 . . 3  |-  ( ph  ->  `' F  e.  _V )
43 coexg 5379 . . 3  |-  ( ( ( F  o.  N
)  e.  _V  /\  `' F  e.  _V )  ->  ( ( F  o.  N )  o.  `' F )  e.  _V )
4440, 42, 43syl2anc 643 . 2  |-  ( ph  ->  ( ( F  o.  N )  o.  `' F )  e.  _V )
45 imasle.l . 2  |-  .<_  =  ( le `  U )
4624, 26, 27, 30, 44, 45strfv3 13465 1  |-  ( ph  -> 
.<_  =  ( ( F  o.  N )  o.  `' F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2924    u. cun 3286   {csn 3782   {cpr 3783   {ctp 3784   <.cop 3785   `'ccnv 4844    o. ccom 4849   -->wf 5417   -onto->wfo 5419   ` cfv 5421  (class class class)co 6048   1c1 8955   2c2 10013  ;cdc 10346   ndxcnx 13429   Basecbs 13432   +g cplusg 13492   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496  TopSetcts 13498   lecple 13499   distcds 13501   TopOpenctopn 13612    "s cimas 13693
This theorem is referenced by:  imasless  13728  imasleval  13729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-imas 13697
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