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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 31331 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 37809 | . . 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 37797 | . . 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 37797 | . . 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 37811 | . . 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 37796 | . . 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 37796 | . . 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 37791 | . 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 5106 βcfv 6497 Basecbs 17088 lecple 17145 Posetcpo 18201 0.cp0 18317 β ccvr 37770 Atomscatm 37771 AtLatcal 37772 |
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-proset 18189 df-poset 18207 df-plt 18224 df-glb 18241 df-p0 18319 df-lat 18326 df-covers 37774 df-ats 37775 df-atl 37806 |
This theorem is referenced by: atncmp 37820 atnlt 37821 atnle 37825 cvlsupr2 37851 cvratlem 37930 2atjm 37954 atbtwn 37955 2atm 38036 2llnmeqat 38080 dalem25 38207 dalem55 38236 dalem57 38238 snatpsubN 38259 pmapat 38272 2llnma1b 38295 cdlemblem 38302 lhp2at0nle 38544 lhpat3 38555 4atexlemcnd 38581 trlval3 38696 cdlemc5 38704 cdleme3 38746 cdleme7 38758 cdleme11k 38777 cdleme16b 38788 cdleme16e 38791 cdleme16f 38792 cdlemednpq 38808 cdleme20j 38827 cdleme22aa 38848 cdleme22cN 38851 cdleme22d 38852 cdlemf2 39071 cdlemb3 39115 cdlemg12e 39156 cdlemg17dALTN 39173 cdlemg19a 39192 cdlemg27b 39205 cdlemg31d 39209 trlcone 39237 cdlemi 39329 tendotr 39339 cdlemk17 39367 cdlemk52 39463 cdleml1N 39485 dia2dimlem1 39573 dia2dimlem2 39574 dia2dimlem3 39575 |
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