Theorem reldmprds 13350

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 13349	. 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. ( ( 2nd ` a ) h c ) , 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 6132	1 |- Rel dom X_s

Syntax hints:   /\ wa 104   A.wral 2510   _Vcvv 2802   [_csb 3127   u. cun 3198   C_ wss 3200   {csn 3669   {cpr 3670   {ctp 3671   <.cop 3672   class class class wbr 4088   {copab 4149   |-> cmpt 4150   X. cxp 4723   dom cdm 4725   ran crn 4726   o. ccom 4729   Rel wrel 4730   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   1stc1st 6300   2ndc2nd 6301   X_cixp 6866   supcsup 7180   0cc0 8031   RR*cxr 8212   < clt 8213   ndxcnx 13078   Basecbs 13081   +g cplusg 13159   .rcmulr 13160  Scalarcsca 13162   .scvsca 13163   .icip 13164  TopSetcts 13165   lecple 13166   distcds 13168   Hom chom 13170  compcco 13171   TopOpenctopn 13322   Xt_cpt 13337   gsum_g cgsu 13339   X__scprds 13347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-dm 4735  df-oprab 6021  df-mpo 6022  df-prds 13349

This theorem is referenced by: (None)