![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > reldmprds | GIF version |
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.) |
Ref | Expression |
---|---|
reldmprds | β’ Rel dom Xs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-prds 12721 | . 2 β’ Xs = (π β V, π β V β¦ β¦Xπ₯ β dom π(Baseβ(πβπ₯)) / π£β¦β¦(π β π£, π β π£ β¦ Xπ₯ β dom π((πβπ₯)(Hom β(πβπ₯))(πβπ₯))) / ββ¦(({β¨(Baseβndx), π£β©, β¨(+gβndx), (π β π£, π β π£ β¦ (π₯ β dom π β¦ ((πβπ₯)(+gβ(πβπ₯))(πβπ₯))))β©, β¨(.rβndx), (π β π£, π β π£ β¦ (π₯ β dom π β¦ ((πβπ₯)(.rβ(πβπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx), π β©, β¨( Β·π βndx), (π β (Baseβπ ), π β π£ β¦ (π₯ β dom π β¦ (π( Β·π β(πβπ₯))(πβπ₯))))β©, β¨(Β·πβndx), (π β π£, π β π£ β¦ (π Ξ£g (π₯ β dom π β¦ ((πβπ₯)(Β·πβ(πβπ₯))(πβπ₯)))))β©}) βͺ ({β¨(TopSetβndx), (βtβ(TopOpen β π))β©, β¨(leβndx), {β¨π, πβ© β£ ({π, π} β π£ β§ βπ₯ β dom π(πβπ₯)(leβ(πβπ₯))(πβπ₯))}β©, β¨(distβndx), (π β π£, π β π£ β¦ sup((ran (π₯ β dom π β¦ ((πβπ₯)(distβ(πβπ₯))(πβπ₯))) βͺ {0}), β*, < ))β©} βͺ {β¨(Hom βndx), ββ©, β¨(compβndx), (π β (π£ Γ π£), π β π£ β¦ (π β (πβ(2nd βπ)), π β (ββπ) β¦ (π₯ β dom π β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((2nd βπ)βπ₯)β©(compβ(πβπ₯))(πβπ₯))(πβπ₯)))))β©}))) | |
2 | 1 | reldmmpo 5988 | 1 β’ Rel dom Xs |
Colors of variables: wff set class |
Syntax hints: β§ wa 104 βwral 2455 Vcvv 2739 β¦csb 3059 βͺ cun 3129 β wss 3131 {csn 3594 {cpr 3595 {ctp 3596 β¨cop 3597 class class class wbr 4005 {copab 4065 β¦ cmpt 4066 Γ cxp 4626 dom cdm 4628 ran crn 4629 β ccom 4632 Rel wrel 4633 βcfv 5218 (class class class)co 5877 β cmpo 5879 1st c1st 6141 2nd c2nd 6142 Xcixp 6700 supcsup 6983 0cc0 7813 β*cxr 7993 < clt 7994 ndxcnx 12461 Basecbs 12464 +gcplusg 12538 .rcmulr 12539 Scalarcsca 12541 Β·π cvsca 12542 Β·πcip 12543 TopSetcts 12544 lecple 12545 distcds 12547 Hom chom 12549 compcco 12550 TopOpenctopn 12694 βtcpt 12709 Ξ£g cgsu 12711 Xscprds 12719 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-br 4006 df-opab 4067 df-xp 4634 df-rel 4635 df-dm 4638 df-oprab 5881 df-mpo 5882 df-prds 12721 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |