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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhplt | Structured version Visualization version GIF version |
Description: An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) |
Ref | Expression |
---|---|
lhplt.l | β’ β€ = (leβπΎ) |
lhplt.s | β’ < = (ltβπΎ) |
lhplt.a | β’ π΄ = (AtomsβπΎ) |
lhplt.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhplt | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β π < π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 766 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β πΎ β HL) | |
2 | simprl 770 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β π β π΄) | |
3 | eqid 2738 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | lhplt.h | . . . 4 β’ π» = (LHypβπΎ) | |
5 | 3, 4 | lhpbase 38347 | . . 3 β’ (π β π» β π β (BaseβπΎ)) |
6 | 5 | ad2antlr 726 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β π β (BaseβπΎ)) |
7 | eqid 2738 | . . . 4 β’ (1.βπΎ) = (1.βπΎ) | |
8 | eqid 2738 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | 7, 8, 4 | lhp1cvr 38348 | . . 3 β’ ((πΎ β HL β§ π β π») β π( β βπΎ)(1.βπΎ)) |
10 | 9 | adantr 482 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β π( β βπΎ)(1.βπΎ)) |
11 | simprr 772 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β€ π)) β π β€ π) | |
12 | lhplt.l | . . 3 β’ β€ = (leβπΎ) | |
13 | lhplt.s | . . 3 β’ < = (ltβπΎ) | |
14 | lhplt.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
15 | 3, 12, 13, 7, 8, 14 | 1cvratlt 37823 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β (BaseβπΎ)) β§ (π( β βπΎ)(1.βπΎ) β§ π β€ π)) β π < π) |
16 | 1, 2, 6, 10, 11, 15 | syl32anc 1379 | 1 β’ (((πΎ β HL 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§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5104 βcfv 6492 Basecbs 17018 lecple 17075 ltcplt 18132 1.cp1 18248 β ccvr 37610 Atomscatm 37611 HLchlt 37698 LHypclh 38333 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-proset 18119 df-poset 18137 df-plt 18154 df-lub 18170 df-glb 18171 df-join 18172 df-meet 18173 df-p0 18249 df-p1 18250 df-lat 18256 df-clat 18323 df-oposet 37524 df-ol 37526 df-oml 37527 df-covers 37614 df-ats 37615 df-atl 37646 df-cvlat 37670 df-hlat 37699 df-lhyp 38337 |
This theorem is referenced by: lhpexle1 38357 |
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