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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 38742 | . 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 38720 | . 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 1160 | 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 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6537 (class class class)co 7405 lecple 17213 joincjn 18276 Atomscatm 38646 CvLatclc 38648 HLchlt 38733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 |
This theorem is referenced by: 2llnneN 38793 atexchcvrN 38824 atbtwnex 38832 3dimlem3 38845 3dimlem3OLDN 38846 3dimlem4 38848 3dimlem4OLDN 38849 hlatexch4 38865 3atlem5 38871 dalem27 39083 cdlemblem 39177 paddasslem1 39204 paddasslem6 39209 cdleme3g 39618 cdleme3h 39619 cdleme7d 39630 cdleme11c 39645 cdleme11dN 39646 cdleme36a 39844 cdlemeg46rgv 39912 cdlemk14 40238 dia2dimlem1 40448 dia2dimlem2 40449 dia2dimlem3 40450 |
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