Theorem reldmprds 12582

Description: The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)

Assertion

Ref	Expression
reldmprds	|- Rel dom X_s

Proof of Theorem reldmprds

Dummy variables a c d e f g h s r x v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prds 12581	. 2 |- X_s = ( s e. _V , r e. _V |-> [_ X_ x e. dom r ( Base ` ( r ` x ) ) / v ]_ [_ ( f e. v , g e. v |-> X_ x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) ) / h ]_ ( ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( +g ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .r ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( .r ` ( r ` x ) ) ( g ` x ) ) ) ) >. } u. { <. (Scalar ` ndx ) , s >. , <. ( .s ` ndx ) , ( f e. ( Base ` s ) , g e. v |-> ( x e. dom r |-> ( f ( .s ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .i ` ndx ) , ( f e. v , g e. v |-> ( s gsumg ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) >. } ) u. ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. r ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ v /\ A. x e. dom r ( f ` x ) ( le ` ( r ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. v , g e. v |-> sup ( ( ran ( x e. dom r |-> ( ( f ` x ) ( dist ` ( r ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( a e. ( v X. v ) , c e. v |-> ( d e. ( c h ( 2nd ` a ) ) , e e. ( h ` a ) |-> ( x e. dom r |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( r ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) ) )
2	1	reldmmpo 5953	1 |- Rel dom X_s

Syntax hints:   /\ wa 103   A.wral 2444   _Vcvv 2726   [_csb 3045   u. cun 3114   C_ wss 3116   {csn 3576   {cpr 3577   {ctp 3578   <.cop 3579   class class class wbr 3982   {copab 4042   |-> cmpt 4043   X. cxp 4602   dom cdm 4604   ran crn 4605   o. ccom 4608   Rel wrel 4609   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107   X_cixp 6664   supcsup 6947   0cc0 7753   RR*cxr 7932   < clt 7933   ndxcnx 12391   Basecbs 12394   +g cplusg 12457   .rcmulr 12458  Scalarcsca 12460   .scvsca 12461   .icip 12462  TopSetcts 12463   lecple 12464   distcds 12466   Hom chom 12468  compcco 12469   TopOpenctopn 12557   Xt_cpt 12572   gsum_g cgsu 12574   X__scprds 12579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-dm 4614  df-oprab 5846  df-mpo 5847  df-prds 12581

This theorem is referenced by: (None)