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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrle | Structured version Visualization version GIF version |
Description: The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
cvrle.b | β’ π΅ = (BaseβπΎ) |
cvrle.l | β’ β€ = (leβπΎ) |
cvrle.c | β’ πΆ = ( β βπΎ) |
Ref | Expression |
---|---|
cvrle | β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ ππΆπ) β π β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrle.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2733 | . . 3 β’ (ltβπΎ) = (ltβπΎ) | |
3 | cvrle.c | . . 3 β’ πΆ = ( β βπΎ) | |
4 | 1, 2, 3 | cvrlt 37778 | . 2 β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ ππΆπ) β π(ltβπΎ)π) |
5 | cvrle.l | . . . 4 β’ β€ = (leβπΎ) | |
6 | 5, 2 | pltval 18226 | . . 3 β’ ((πΎ β π΄ β§ π β π΅ β§ π β π΅) β (π(ltβπΎ)π β (π β€ π β§ π β π))) |
7 | 6 | simprbda 500 | . 2 β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ π(ltβπΎ)π) β π β€ π) |
8 | 4, 7 | syldan 592 | 1 β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ ππΆπ) β π β€ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 class class class wbr 5106 βcfv 6497 Basecbs 17088 lecple 17145 ltcplt 18202 β ccvr 37770 |
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-pow 5321 ax-pr 5385 ax-un 7673 |
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-ne 2941 df-ral 3062 df-rex 3071 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-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-plt 18224 df-covers 37774 |
This theorem is referenced by: cvrnbtwn4 37787 cvrcmp 37791 atcvrj2b 37941 atexchcvrN 37949 llncmp 38031 llncvrlpln 38067 lplncmp 38071 lplncvrlvol 38125 lvolcmp 38126 |
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