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Mirrors > Home > MPE Home > Th. List > reldmprds | Structured version Visualization version 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.) (Revised by Zhi Wang, 18-Aug-2024.) |
Ref | Expression |
---|---|
reldmprds | β’ Rel dom Xs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-prds 17334 | . 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 ((2nd βπ)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 7491 | 1 β’ Rel dom Xs |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 βwral 3061 Vcvv 3444 β¦csb 3856 βͺ cun 3909 β wss 3911 {csn 4587 {cpr 4589 {ctp 4591 β¨cop 4593 class class class wbr 5106 {copab 5168 β¦ cmpt 5189 Γ cxp 5632 dom cdm 5634 ran crn 5635 β ccom 5638 Rel wrel 5639 βcfv 6497 (class class class)co 7358 β cmpo 7360 1st c1st 7920 2nd c2nd 7921 Xcixp 8838 supcsup 9381 0cc0 11056 β*cxr 11193 < clt 11194 ndxcnx 17070 Basecbs 17088 +gcplusg 17138 .rcmulr 17139 Scalarcsca 17141 Β·π cvsca 17142 Β·πcip 17143 TopSetcts 17144 lecple 17145 distcds 17147 Hom chom 17149 compcco 17150 TopOpenctopn 17308 βtcpt 17325 Ξ£g cgsu 17327 Xscprds 17332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-dm 5644 df-oprab 7362 df-mpo 7363 df-prds 17334 |
This theorem is referenced by: dsmmval 21156 dsmmval2 21158 dsmmbas2 21159 dsmmfi 21160 |
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