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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > llni | Structured version Visualization version GIF version |
Description: Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.) |
Ref | Expression |
---|---|
llnset.b | β’ π΅ = (BaseβπΎ) |
llnset.c | β’ πΆ = ( β βπΎ) |
llnset.a | β’ π΄ = (AtomsβπΎ) |
llnset.n | β’ π = (LLinesβπΎ) |
Ref | Expression |
---|---|
llni | β’ (((πΎ β π· β§ π β π΅ β§ π β π΄) β§ ππΆπ) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 1190 | . 2 β’ (((πΎ β π· β§ π β π΅ β§ π β π΄) β§ ππΆπ) β π β π΅) | |
2 | breq1 5146 | . . . 4 β’ (π = π β (ππΆπ β ππΆπ)) | |
3 | 2 | rspcev 3608 | . . 3 β’ ((π β π΄ β§ ππΆπ) β βπ β π΄ ππΆπ) |
4 | 3 | 3ad2antl3 1185 | . 2 β’ (((πΎ β π· β§ π β π΅ β§ π β π΄) β§ ππΆπ) β βπ β π΄ ππΆπ) |
5 | simpl1 1189 | . . 3 β’ (((πΎ β π· β§ π β π΅ β§ π β π΄) β§ ππΆπ) β πΎ β π·) | |
6 | llnset.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
7 | llnset.c | . . . 4 β’ πΆ = ( β βπΎ) | |
8 | llnset.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
9 | llnset.n | . . . 4 β’ π = (LLinesβπΎ) | |
10 | 6, 7, 8, 9 | islln 38974 | . . 3 β’ (πΎ β π· β (π β π β (π β π΅ β§ βπ β π΄ ππΆπ))) |
11 | 5, 10 | syl 17 | . 2 β’ (((πΎ β π· β§ π β π΅ β§ π β π΄) β§ ππΆπ) β (π β π β (π β π΅ β§ βπ β π΄ ππΆπ))) |
12 | 1, 4, 11 | mpbir2and 712 | 1 β’ (((πΎ β π· β§ π β π΅ β§ π β π΄) β§ ππΆπ) β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βwrex 3066 class class class wbr 5143 βcfv 6543 Basecbs 17174 β ccvr 38729 Atomscatm 38730 LLinesclln 38959 |
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-sep 5294 ax-nul 5301 ax-pr 5424 |
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-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-llines 38966 |
This theorem is referenced by: llnle 38986 atcvrlln 38988 lplncvrlvol 39084 |
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