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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leat2 | Structured version Visualization version GIF version |
Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.) |
Ref | Expression |
---|---|
leatom.b | β’ π΅ = (BaseβπΎ) |
leatom.l | β’ β€ = (leβπΎ) |
leatom.z | β’ 0 = (0.βπΎ) |
leatom.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
leat2 | β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ (π β 0 β§ π β€ π)) β π = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leatom.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
2 | leatom.l | . . . . . 6 β’ β€ = (leβπΎ) | |
3 | leatom.z | . . . . . 6 β’ 0 = (0.βπΎ) | |
4 | leatom.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
5 | 1, 2, 3, 4 | leatb 38758 | . . . . 5 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β€ π β (π = π β¨ π = 0 ))) |
6 | orcom 869 | . . . . . 6 β’ ((π = π β¨ π = 0 ) β (π = 0 β¨ π = π)) | |
7 | neor 3030 | . . . . . 6 β’ ((π = 0 β¨ π = π) β (π β 0 β π = π)) | |
8 | 6, 7 | bitri 275 | . . . . 5 β’ ((π = π β¨ π = 0 ) β (π β 0 β π = π)) |
9 | 5, 8 | bitrdi 287 | . . . 4 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β€ π β (π β 0 β π = π))) |
10 | 9 | biimpd 228 | . . 3 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β€ π β (π β 0 β π = π))) |
11 | 10 | com23 86 | . 2 β’ ((πΎ β OP β§ π β π΅ β§ π β π΄) β (π β 0 β (π β€ π β π = π))) |
12 | 11 | imp32 418 | 1 β’ (((πΎ β OP β§ π β π΅ β§ π β π΄) β§ (π β 0 β§ π β€ π)) β π = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β¨ wo 846 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2936 class class class wbr 5142 βcfv 6542 Basecbs 17173 lecple 17233 0.cp0 18408 OPcops 38638 Atomscatm 38729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-proset 18280 df-poset 18298 df-plt 18315 df-glb 18332 df-p0 18410 df-oposet 38642 df-covers 38732 df-ats 38733 |
This theorem is referenced by: dalemcea 39127 cdlemg12g 40116 |
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