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Mirrors > Home > MPE Home > Th. List > Mathboxes > relrngo | Structured version Visualization version GIF version |
Description: The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
relrngo | β’ Rel RingOps |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rngo 36404 | . 2 β’ RingOps = {β¨π, ββ© β£ ((π β AbelOp β§ β:(ran π Γ ran π)βΆran π) β§ (βπ₯ β ran πβπ¦ β ran πβπ§ β ran π(((π₯βπ¦)βπ§) = (π₯β(π¦βπ§)) β§ (π₯β(π¦ππ§)) = ((π₯βπ¦)π(π₯βπ§)) β§ ((π₯ππ¦)βπ§) = ((π₯βπ§)π(π¦βπ§))) β§ βπ₯ β ran πβπ¦ β ran π((π₯βπ¦) = π¦ β§ (π¦βπ₯) = π¦)))} | |
2 | 1 | relopabiv 5780 | 1 β’ Rel RingOps |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 Γ cxp 5635 ran crn 5638 Rel wrel 5642 βΆwf 6496 (class class class)co 7361 AbelOpcablo 29535 RingOpscrngo 36403 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-in 3921 df-ss 3931 df-opab 5172 df-xp 5643 df-rel 5644 df-rngo 36404 |
This theorem is referenced by: isrngo 36406 rngoi 36408 rngoablo2 36418 rngosn3 36433 isdrngo1 36465 iscrngo2 36506 |
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