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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islln2a | Structured version Visualization version GIF version |
Description: The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
islln2a.j | β’ β¨ = (joinβπΎ) |
islln2a.a | β’ π΄ = (AtomsβπΎ) |
islln2a.n | β’ π = (LLinesβπΎ) |
Ref | Expression |
---|---|
islln2a | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β ((π β¨ π) β π β π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7431 | . . . . . 6 β’ (π = π β (π β¨ π) = (π β¨ π)) | |
2 | islln2a.j | . . . . . . . 8 β’ β¨ = (joinβπΎ) | |
3 | islln2a.a | . . . . . . . 8 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | hlatjidm 38845 | . . . . . . 7 β’ ((πΎ β HL β§ π β π΄) β (π β¨ π) = π) |
5 | 4 | 3adant2 1128 | . . . . . 6 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) = π) |
6 | 1, 5 | sylan9eqr 2789 | . . . . 5 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π = π) β (π β¨ π) = π) |
7 | islln2a.n | . . . . . . . . . . 11 β’ π = (LLinesβπΎ) | |
8 | 3, 7 | llnneat 38991 | . . . . . . . . . 10 β’ ((πΎ β HL β§ π β π) β Β¬ π β π΄) |
9 | 8 | adantlr 713 | . . . . . . . . 9 β’ (((πΎ β HL β§ π β π΄) β§ π β π) β Β¬ π β π΄) |
10 | 9 | ex 411 | . . . . . . . 8 β’ ((πΎ β HL β§ π β π΄) β (π β π β Β¬ π β π΄)) |
11 | 10 | con2d 134 | . . . . . . 7 β’ ((πΎ β HL β§ π β π΄) β (π β π΄ β Β¬ π β π)) |
12 | 11 | 3impia 1114 | . . . . . 6 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β Β¬ π β π) |
13 | 12 | adantr 479 | . . . . 5 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π = π) β Β¬ π β π) |
14 | 6, 13 | eqneltrd 2848 | . . . 4 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π = π) β Β¬ (π β¨ π) β π) |
15 | 14 | ex 411 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π = π β Β¬ (π β¨ π) β π)) |
16 | 15 | necon2ad 2951 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β ((π β¨ π) β π β π β π)) |
17 | 2, 3, 7 | llni2 38989 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β π) β (π β¨ π) β π) |
18 | 17 | ex 411 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β π β (π β¨ π) β π)) |
19 | 16, 18 | impbid 211 | 1 β’ ((πΎ β HL 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¬ wn 3 β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2936 βcfv 6551 (class class class)co 7424 joincjn 18308 Atomscatm 38739 HLchlt 38826 LLinesclln 38968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-lat 18429 df-clat 18496 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-llines 38975 |
This theorem is referenced by: cdleme16d 39758 |
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