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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatexch1 | Structured version Visualization version GIF version |
Description: Atom exchange property. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlatexch.l | β’ β€ = (leβπΎ) |
cvlatexch.j | β’ β¨ = (joinβπΎ) |
cvlatexch.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
cvlatexch1 | 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))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatexch.l | . . 3 β’ β€ = (leβπΎ) | |
2 | cvlatexch.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | cvlatexch.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | 1, 2, 3 | cvlatexchb1 37842 | . 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) = (π β¨ π))) |
5 | cvllat 37834 | . . . . 5 β’ (πΎ β CvLat β πΎ β Lat) | |
6 | 5 | 3ad2ant1 1134 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β πΎ β Lat) |
7 | simp23 1209 | . . . . 5 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β π β π΄) | |
8 | eqid 2733 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
9 | 8, 3 | atbase 37797 | . . . . 5 β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 7, 9 | syl 17 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β π β (BaseβπΎ)) |
11 | simp22 1208 | . . . . 5 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β π β π΄) | |
12 | 8, 3 | atbase 37797 | . . . . 5 β’ (π β π΄ β π β (BaseβπΎ)) |
13 | 11, 12 | syl 17 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π ) β π β (BaseβπΎ)) |
14 | 8, 1, 2 | latlej2 18343 | . . . 4 β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β π β€ (π β¨ π)) |
15 | 6, 10, 13, 14 | syl3anc 1372 | . . 3 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)) |
16 | breq2 5110 | . . 3 β’ ((π β¨ π) = (π β¨ π) β (π β€ (π β¨ π) β π β€ (π β¨ π))) | |
17 | 15, 16 | syl5ibrcom 247 | . 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) = (π β¨ π) β π β€ (π β¨ π))) |
18 | 4, 17 | sylbid 239 | 1 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))) |
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 Basecbs 17088 lecple 17145 joincjn 18205 Latclat 18325 Atomscatm 37771 CvLatclc 37773 |
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 |
This theorem is referenced by: cvlatexch2 37845 cvlsupr2 37851 hlatexch1 37904 4atex 38585 cdleme20zN 38810 cdleme19a 38812 cdleme21b 38835 cdleme21c 38836 cdleme22g 38857 cdlemf1 39070 |
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