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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atcmp | Structured version Visualization version GIF version |
Description: If two atoms are comparable, they are equal. (atsseq 31631 analog.) (Contributed by NM, 13-Oct-2011.) |
Ref | Expression |
---|---|
atcmp.l | β’ β€ = (leβπΎ) |
atcmp.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atcmp | β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β (π β€ π β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atlpos 38219 | . . 3 β’ (πΎ β AtLat β πΎ β Poset) | |
2 | 1 | 3ad2ant1 1134 | . 2 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β πΎ β Poset) |
3 | eqid 2733 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | atcmp.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | atbase 38207 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
6 | 5 | 3ad2ant2 1135 | . 2 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β π β (BaseβπΎ)) |
7 | 3, 4 | atbase 38207 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
8 | 7 | 3ad2ant3 1136 | . 2 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β π β (BaseβπΎ)) |
9 | eqid 2733 | . . . 4 β’ (0.βπΎ) = (0.βπΎ) | |
10 | 3, 9 | atl0cl 38221 | . . 3 β’ (πΎ β AtLat β (0.βπΎ) β (BaseβπΎ)) |
11 | 10 | 3ad2ant1 1134 | . 2 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β (0.βπΎ) β (BaseβπΎ)) |
12 | eqid 2733 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
13 | 9, 12, 4 | atcvr0 38206 | . . 3 β’ ((πΎ β AtLat β§ π β π΄) β (0.βπΎ)( β βπΎ)π) |
14 | 13 | 3adant3 1133 | . 2 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β (0.βπΎ)( β βπΎ)π) |
15 | 9, 12, 4 | atcvr0 38206 | . . 3 β’ ((πΎ β AtLat β§ π β π΄) β (0.βπΎ)( β βπΎ)π) |
16 | 15 | 3adant2 1132 | . 2 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β (0.βπΎ)( β βπΎ)π) |
17 | atcmp.l | . . 3 β’ β€ = (leβπΎ) | |
18 | 3, 17, 12 | cvrcmp 38201 | . 2 β’ ((πΎ β Poset β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ (0.βπΎ) β (BaseβπΎ)) β§ ((0.βπΎ)( β βπΎ)π β§ (0.βπΎ)( β βπΎ)π)) β (π β€ π β π = π)) |
19 | 2, 6, 8, 11, 14, 16, 18 | syl132anc 1389 | 1 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β (π β€ π β π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 Basecbs 17144 lecple 17204 Posetcpo 18260 0.cp0 18376 β ccvr 38180 Atomscatm 38181 AtLatcal 38182 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-proset 18248 df-poset 18266 df-plt 18283 df-glb 18300 df-p0 18378 df-lat 18385 df-covers 38184 df-ats 38185 df-atl 38216 |
This theorem is referenced by: atncmp 38230 atnlt 38231 atnle 38235 cvlsupr2 38261 cvratlem 38340 2atjm 38364 atbtwn 38365 2atm 38446 2llnmeqat 38490 dalem25 38617 dalem55 38646 dalem57 38648 snatpsubN 38669 pmapat 38682 2llnma1b 38705 cdlemblem 38712 lhp2at0nle 38954 lhpat3 38965 4atexlemcnd 38991 trlval3 39106 cdlemc5 39114 cdleme3 39156 cdleme7 39168 cdleme11k 39187 cdleme16b 39198 cdleme16e 39201 cdleme16f 39202 cdlemednpq 39218 cdleme20j 39237 cdleme22aa 39258 cdleme22cN 39261 cdleme22d 39262 cdlemf2 39481 cdlemb3 39525 cdlemg12e 39566 cdlemg17dALTN 39583 cdlemg19a 39602 cdlemg27b 39615 cdlemg31d 39619 trlcone 39647 cdlemi 39739 tendotr 39749 cdlemk17 39777 cdlemk52 39873 cdleml1N 39895 dia2dimlem1 39983 dia2dimlem2 39984 dia2dimlem3 39985 |
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