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Theorem prdsco 14418
Description: Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
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 2443 . . 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 14407 . . 3  |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
8 eqid 2443 . . . 4  |-  ( +g  `  P )  =  ( +g  `  P )
91, 4, 5, 6, 3, 8prdsplusg 14408 . . 3  |-  ( ph  ->  ( +g  `  P
)  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
10 eqid 2443 . . . 4  |-  ( .r
`  P )  =  ( .r `  P
)
111, 4, 5, 6, 3, 10prdsmulr 14409 . . 3  |-  ( ph  ->  ( .r `  P
)  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r `  ( R `  x )
) ( g `  x ) ) ) ) )
12 eqid 2443 . . . 4  |-  ( .s
`  P )  =  ( .s `  P
)
131, 4, 5, 6, 3, 2, 12prdsvsca 14410 . . 3  |-  ( ph  ->  ( .s `  P
)  =  ( f  e.  ( Base `  S
) ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `
 x ) ) ( g `  x
) ) ) ) )
14 eqidd 2444 . . 3  |-  ( ph  ->  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i `  ( R `  x )
) ( g `  x ) ) ) ) ) )
15 eqid 2443 . . . 4  |-  (TopSet `  P )  =  (TopSet `  P )
161, 4, 5, 6, 3, 15prdstset 14416 . . 3  |-  ( ph  ->  (TopSet `  P )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
17 eqid 2443 . . . 4  |-  ( le
`  P )  =  ( le `  P
)
181, 4, 5, 6, 3, 17prdsle 14412 . . 3  |-  ( ph  ->  ( le `  P
)  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
19 eqid 2443 . . . 4  |-  ( dist `  P )  =  (
dist `  P )
201, 4, 5, 6, 3, 19prdsds 14414 . . 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* ,  <  ) ) )
21 prdshom.h . . . 4  |-  H  =  ( Hom  `  P
)
221, 4, 5, 6, 3, 21prdshom 14417 . . 3  |-  ( ph  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) )
23 eqidd 2444 . . 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 ) ) ) ) ) )
241, 2, 3, 7, 9, 11, 13, 14, 16, 18, 20, 22, 23, 4, 5prdsval 14405 . 2  |-  ( ph  ->  P  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  P ) >. ,  <. ( .r `  ndx ) ,  ( .r `  P ) >. }  u.  {
<. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( .s `  P
) >. ,  <. ( .i `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )
>. } )  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 ) ) ) ) )
>. } ) ) )
25 prdsco.o . 2  |-  .xb  =  (comp `  P )
26 ccoid 14368 . 2  |- comp  = Slot  (comp ` 
ndx )
27 fvex 5713 . . . . . 6  |-  ( Base `  P )  e.  _V
286, 27eqeltri 2513 . . . . 5  |-  B  e. 
_V
2928, 28xpex 6520 . . . 4  |-  ( B  X.  B )  e. 
_V
3029, 28mpt2ex 6662 . . 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
3130a1i 11 . 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 )
32 snsspr2 4035 . . . 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 ) ) ) ) )
>. }
33 ssun2 3532 . . . 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 ) ) ) ) )
>. } )
3432, 33sstri 3377 . . 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 ) ) ) ) )
>. } )
35 ssun2 3532 . . 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
) >. ,  <. ( .i `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )
>. } )  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 ) ) ) ) )
>. } ) )
3634, 35sstri 3377 . 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
) >. ,  <. ( .i `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )
>. } )  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 ) ) ) ) )
>. } ) )
3724, 25, 26, 31, 36prdsvallem 14404 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2984    u. cun 3338   {csn 3889   {cpr 3891   {ctp 3893   <.cop 3895    e. cmpt 4362    X. cxp 4850   dom cdm 4852   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   1stc1st 6587   2ndc2nd 6588   ndxcnx 14183   Basecbs 14186   +g cplusg 14250   .rcmulr 14251  Scalarcsca 14253   .scvsca 14254   .icip 14255  TopSetcts 14256   lecple 14257   distcds 14259   Hom chom 14261  compcco 14262    gsumg cgsu 14391   X_scprds 14396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-hom 14274  df-cco 14275  df-prds 14398
This theorem is referenced by: (None)
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