MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasle Structured version   Unicode version

Theorem imasle 14781
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 2467 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2467 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2467 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2467 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2467 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2467 . . 3  |-  ( .i
`  R )  =  ( .i `  R
)
9 eqid 2467 . . 3  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
10 eqid 2467 . . 3  |-  ( dist `  R )  =  (
dist `  R )
11 imasle.n . . 3  |-  N  =  ( le `  R
)
12 imasbas.f . . . 4  |-  ( ph  ->  F : V -onto-> B
)
13 imasbas.r . . . 4  |-  ( ph  ->  R  e.  Z )
14 eqid 2467 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
151, 2, 12, 13, 3, 14imasplusg 14775 . . 3  |-  ( ph  ->  ( +g  `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
16 eqid 2467 . . . 4  |-  ( .r
`  U )  =  ( .r `  U
)
171, 2, 12, 13, 4, 16imasmulr 14776 . . 3  |-  ( ph  ->  ( .r `  U
)  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( .r `  R
) q ) )
>. } )
18 eqid 2467 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
191, 2, 12, 13, 5, 6, 7, 18imasvsca 14778 . . 3  |-  ( ph  ->  ( .s `  U
)  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
20 eqidd 2468 . . 3  |-  ( 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 eqid 2467 . . . 4  |-  (TopSet `  U )  =  (TopSet `  U )
221, 2, 12, 13, 9, 21imastset 14780 . . 3  |-  ( ph  ->  (TopSet `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
23 eqid 2467 . . . 4  |-  ( dist `  U )  =  (
dist `  U )
241, 2, 12, 13, 10, 23imasds 14771 . . 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* ,  `'  <  ) ) )
25 eqidd 2468 . . 3  |-  ( ph  ->  ( ( F  o.  N )  o.  `' F )  =  ( ( F  o.  N
)  o.  `' F
) )
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 20, 22, 24, 25, 12, 13imasval 14769 . 2  |-  ( 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 ) ,  (TopSet `  U
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } ) )
27 eqid 2467 . . 3  |-  ( ( { <. ( 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 ) ,  (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 ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  (TopSet `  U
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } )
2827imasvalstr 14710 . 2  |-  ( ( { <. ( 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 ) ,  (TopSet `  U
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } ) Struct  <. 1 , ; 1 2 >.
29 pleid 14653 . 2  |-  le  = Slot  ( le `  ndx )
30 snsstp2 4179 . . 3  |-  { <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. }  C_  {
<. (TopSet `  ndx ) ,  (TopSet `  U ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F )
>. ,  <. ( dist `  ndx ) ,  (
dist `  U ) >. }
31 ssun2 3668 . . 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 ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  (TopSet `  U
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } )
3230, 31sstri 3513 . 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 ) >. ,  <. ( .i `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  (TopSet `  U
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  N )  o.  `' F ) >. ,  <. (
dist `  ndx ) ,  ( dist `  U
) >. } )
33 fof 5795 . . . . . 6  |-  ( F : V -onto-> B  ->  F : V --> B )
3412, 33syl 16 . . . . 5  |-  ( ph  ->  F : V --> B )
35 fvex 5876 . . . . . 6  |-  ( Base `  R )  e.  _V
362, 35syl6eqel 2563 . . . . 5  |-  ( ph  ->  V  e.  _V )
37 fex 6134 . . . . 5  |-  ( ( F : V --> B  /\  V  e.  _V )  ->  F  e.  _V )
3834, 36, 37syl2anc 661 . . . 4  |-  ( ph  ->  F  e.  _V )
39 fvex 5876 . . . . 5  |-  ( le
`  R )  e. 
_V
4011, 39eqeltri 2551 . . . 4  |-  N  e. 
_V
41 coexg 6736 . . . 4  |-  ( ( F  e.  _V  /\  N  e.  _V )  ->  ( F  o.  N
)  e.  _V )
4238, 40, 41sylancl 662 . . 3  |-  ( ph  ->  ( F  o.  N
)  e.  _V )
43 cnvexg 6731 . . . 4  |-  ( F  e.  _V  ->  `' F  e.  _V )
4438, 43syl 16 . . 3  |-  ( ph  ->  `' F  e.  _V )
45 coexg 6736 . . 3  |-  ( ( ( F  o.  N
)  e.  _V  /\  `' F  e.  _V )  ->  ( ( F  o.  N )  o.  `' F )  e.  _V )
4642, 44, 45syl2anc 661 . 2  |-  ( ph  ->  ( ( F  o.  N )  o.  `' F )  e.  _V )
47 imasle.l . 2  |-  .<_  =  ( le `  U )
4826, 28, 29, 32, 46, 47strfv3 14528 1  |-  ( ph  -> 
.<_  =  ( ( F  o.  N )  o.  `' F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474   {csn 4027   {ctp 4031   <.cop 4033   U_ciun 4325   `'ccnv 4998    o. ccom 5003   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6285   1c1 9494   2c2 10586  ;cdc 10977   ndxcnx 14490   Basecbs 14493   +g cplusg 14558   .rcmulr 14559  Scalarcsca 14561   .scvsca 14562   .icip 14563  TopSetcts 14564   lecple 14565   distcds 14567   TopOpenctopn 14680    "s cimas 14762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-imas 14766
This theorem is referenced by:  imasless  14798  imasleval  14799
  Copyright terms: Public domain W3C validator