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Theorem prdsco 13367
Description: Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
prdsbas.p  |-  P  =  ( S X_s R )
prdsbas.s  |-  ( ph  ->  S  e.  V )
prdsbas.r  |-  ( ph  ->  R  e.  W )
prdsbas.b  |-  B  =  ( Base `  P
)
prdsbas.i  |-  ( ph  ->  dom  R  =  I )
prdshom.h  |-  H  =  (  Hom  `  P
)
prdsco.o  |-  .xb  =  (comp `  P )
Assertion
Ref Expression
prdsco  |-  ( ph  -> 
.xb  =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) )
Distinct variable groups:    a, c,
d, e, x, B    H, a, c, d, e    ph, a, c, d, e, x    I, a, c, d, e, x    x, P    R, a, c, d, e, x    S, a, c, d, e, x
Allowed substitution hints:    P( e, a, c, d)    .xb ( x, e, a, c, d)    H( x)    V( x, e, a, c, d)    W( x, e, a, c, d)

Proof of Theorem prdsco
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3  |-  P  =  ( S X_s R )
2 eqid 2283 . . 3  |-  ( Base `  S )  =  (
Base `  S )
3 prdsbas.i . . 3  |-  ( ph  ->  dom  R  =  I )
4 prdsbas.s . . . 4  |-  ( ph  ->  S  e.  V )
5 prdsbas.r . . . 4  |-  ( ph  ->  R  e.  W )
6 prdsbas.b . . . 4  |-  B  =  ( Base `  P
)
71, 4, 5, 6, 3prdsbas 13357 . . 3  |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
8 eqid 2283 . . . 4  |-  ( +g  `  P )  =  ( +g  `  P )
91, 4, 5, 6, 3, 8prdsplusg 13358 . . 3  |-  ( ph  ->  ( +g  `  P
)  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
10 eqid 2283 . . . 4  |-  ( .r
`  P )  =  ( .r `  P
)
111, 4, 5, 6, 3, 10prdsmulr 13359 . . 3  |-  ( ph  ->  ( .r `  P
)  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r `  ( R `  x )
) ( g `  x ) ) ) ) )
12 eqid 2283 . . . 4  |-  ( .s
`  P )  =  ( .s `  P
)
131, 4, 5, 6, 3, 2, 12prdsvsca 13360 . . 3  |-  ( ph  ->  ( .s `  P
)  =  ( f  e.  ( Base `  S
) ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `
 x ) ) ( g `  x
) ) ) ) )
14 eqid 2283 . . . 4  |-  (TopSet `  P )  =  (TopSet `  P )
151, 4, 5, 6, 3, 14prdstset 13365 . . 3  |-  ( ph  ->  (TopSet `  P )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
16 eqid 2283 . . . 4  |-  ( le
`  P )  =  ( le `  P
)
171, 4, 5, 6, 3, 16prdsle 13361 . . 3  |-  ( ph  ->  ( le `  P
)  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
18 eqid 2283 . . . 4  |-  ( dist `  P )  =  (
dist `  P )
191, 4, 5, 6, 3, 18prdsds 13363 . . 3  |-  ( ph  ->  ( dist `  P
)  =  ( f  e.  B ,  g  e.  B  |->  sup (
( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
20 prdshom.h . . . 4  |-  H  =  (  Hom  `  P
)
211, 4, 5, 6, 3, 20prdshom 13366 . . 3  |-  ( ph  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
(  Hom  `  ( R `
 x ) ) ( g `  x
) ) ) )
22 eqidd 2284 . . 3  |-  ( ph  ->  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )  =  ( a  e.  ( B  X.  B
) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `  a
)  |->  ( x  e.  I  |->  ( ( d `
 x ) (
<. ( ( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) ) )
231, 2, 3, 7, 9, 11, 13, 15, 17, 19, 21, 22, 4, 5prdsval 13355 . 2  |-  ( ph  ->  P  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  P ) >. ,  <. ( .r `  ndx ) ,  ( .r `  P ) >. }  u.  {
<. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( .s `  P
) >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  (TopSet `  P ) >. ,  <. ( le `  ndx ) ,  ( le `  P ) >. ,  <. (
dist `  ndx ) ,  ( dist `  P
) >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) ) )
24 prdsco.o . 2  |-  .xb  =  (comp `  P )
25 ccoid 13322 . 2  |- comp  = Slot  (comp ` 
ndx )
26 fvex 5539 . . . . . 6  |-  ( Base `  P )  e.  _V
276, 26eqeltri 2353 . . . . 5  |-  B  e. 
_V
2827, 27xpex 4801 . . . 4  |-  ( B  X.  B )  e. 
_V
2928, 27mpt2ex 6198 . . 3  |-  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) )  e.  _V
3029a1i 10 . 2  |-  ( ph  ->  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )  e.  _V )
31 snsspr2 3765 . . . 4  |-  { <. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. }  C_  {
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. }
32 ssun2 3339 . . . 4  |-  { <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. }  C_  ( { <. (TopSet `  ndx ) ,  (TopSet `  P
) >. ,  <. ( le `  ndx ) ,  ( le `  P
) >. ,  <. ( dist `  ndx ) ,  ( dist `  P
) >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } )
3331, 32sstri 3188 . . 3  |-  { <. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. }  C_  ( { <. (TopSet `  ndx ) ,  (TopSet `  P
) >. ,  <. ( le `  ndx ) ,  ( le `  P
) >. ,  <. ( dist `  ndx ) ,  ( dist `  P
) >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } )
34 ssun2 3339 . . 3  |-  ( {
<. (TopSet `  ndx ) ,  (TopSet `  P ) >. ,  <. ( le `  ndx ) ,  ( le
`  P ) >. ,  <. ( dist `  ndx ) ,  ( dist `  P ) >. }  u.  {
<. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } )  C_  (
( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  P
) >. ,  <. ( .r `  ndx ) ,  ( .r `  P
) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( .s `  P
) >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  (TopSet `  P ) >. ,  <. ( le `  ndx ) ,  ( le `  P ) >. ,  <. (
dist `  ndx ) ,  ( dist `  P
) >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) )
3533, 34sstri 3188 . 2  |-  { <. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. }  C_  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  P
) >. ,  <. ( .r `  ndx ) ,  ( .r `  P
) >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( .s `  P
) >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  (TopSet `  P ) >. ,  <. ( le `  ndx ) ,  ( le `  P ) >. ,  <. (
dist `  ndx ) ,  ( dist `  P
) >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) )
3623, 24, 25, 30, 35prdsvallem 13354 1  |-  ( ph  -> 
.xb  =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   {csn 3640   {cpr 3641   {ctp 3642   <.cop 3643    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  TopSetcts 13214   lecple 13215   distcds 13217    Hom chom 13219  compcco 13220   X_scprds 13346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348
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