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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islpln4 | Structured version Visualization version GIF version |
Description: The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.) |
Ref | Expression |
---|---|
lplnset.b | β’ π΅ = (BaseβπΎ) |
lplnset.c | β’ πΆ = ( β βπΎ) |
lplnset.n | β’ π = (LLinesβπΎ) |
lplnset.p | β’ π = (LPlanesβπΎ) |
Ref | Expression |
---|---|
islpln4 | β’ ((πΎ β π΄ β§ π β π΅) β (π β π β βπ¦ β π π¦πΆπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lplnset.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | lplnset.c | . . 3 β’ πΆ = ( β βπΎ) | |
3 | lplnset.n | . . 3 β’ π = (LLinesβπΎ) | |
4 | lplnset.p | . . 3 β’ π = (LPlanesβπΎ) | |
5 | 1, 2, 3, 4 | islpln 38022 | . 2 β’ (πΎ β π΄ β (π β π β (π β π΅ β§ βπ¦ β π π¦πΆπ))) |
6 | 5 | baibd 541 | 1 β’ ((πΎ β π΄ β§ π β π΅) β (π β π β βπ¦ β π π¦πΆπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3074 class class class wbr 5110 βcfv 6501 Basecbs 17090 β ccvr 37753 LLinesclln 37983 LPlanesclpl 37984 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-lplanes 37991 |
This theorem is referenced by: islpln3 38025 lplncmp 38054 |
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