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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatexch2 | Structured version Visualization version GIF version |
Description: Atom exchange property. (Contributed by NM, 8-Jan-2012.) |
Ref | Expression |
---|---|
hlatexchb.l | β’ β€ = (leβπΎ) |
hlatexchb.j | β’ β¨ = (joinβπΎ) |
hlatexchb.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlatexch2 | β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π β€ (π β¨ π ) β π β€ (π β¨ π ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcvl 37867 | . 2 β’ (πΎ β HL β πΎ β CvLat) | |
2 | hlatexchb.l | . . 3 β’ β€ = (leβπΎ) | |
3 | hlatexchb.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | hlatexchb.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | 2, 3, 4 | cvlatexch2 37845 | . 2 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π β€ (π β¨ π ) β π β€ (π β¨ π ))) |
6 | 1, 5 | syl3an1 1164 | 1 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β (π β€ (π β¨ π ) β π β€ (π β¨ π ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 class class class wbr 5106 βcfv 6497 (class class class)co 7358 lecple 17145 joincjn 18205 Atomscatm 37771 CvLatclc 37773 HLchlt 37858 |
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-rep 5243 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-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 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-iun 4957 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-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-lat 18326 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 |
This theorem is referenced by: 2llnneN 37918 atexchcvrN 37949 atbtwnex 37957 3dimlem3 37970 3dimlem3OLDN 37971 3dimlem4 37973 3dimlem4OLDN 37974 hlatexch4 37990 3atlem5 37996 dalem27 38208 cdlemblem 38302 paddasslem1 38329 paddasslem6 38334 cdleme3g 38743 cdleme3h 38744 cdleme7d 38755 cdleme11c 38770 cdleme11dN 38771 cdleme36a 38969 cdlemeg46rgv 39037 cdlemk14 39363 dia2dimlem1 39573 dia2dimlem2 39574 dia2dimlem3 39575 |
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