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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islpln | Structured version Visualization version GIF version |
Description: The predicate "is a lattice plane". (Contributed by NM, 16-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 |
---|---|
islpln | 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 | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | lplnset.c | . . . 4 β’ πΆ = ( β βπΎ) | |
3 | lplnset.n | . . . 4 β’ π = (LLinesβπΎ) | |
4 | lplnset.p | . . . 4 β’ π = (LPlanesβπΎ) | |
5 | 1, 2, 3, 4 | lplnset 39002 | . . 3 β’ (πΎ β π΄ β π = {π₯ β π΅ β£ βπ¦ β π π¦πΆπ₯}) |
6 | 5 | eleq2d 2815 | . 2 β’ (πΎ β π΄ β (π β π β π β {π₯ β π΅ β£ βπ¦ β π π¦πΆπ₯})) |
7 | breq2 5152 | . . . 4 β’ (π₯ = π β (π¦πΆπ₯ β π¦πΆπ)) | |
8 | 7 | rexbidv 3175 | . . 3 β’ (π₯ = π β (βπ¦ β π π¦πΆπ₯ β βπ¦ β π π¦πΆπ)) |
9 | 8 | elrab 3682 | . 2 β’ (π β {π₯ β π΅ β£ βπ¦ β π π¦πΆπ₯} β (π β π΅ β§ βπ¦ β π π¦πΆπ)) |
10 | 6, 9 | bitrdi 287 | 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 395 = wceq 1534 β wcel 2099 βwrex 3067 {crab 3429 class class class wbr 5148 βcfv 6548 Basecbs 17180 β ccvr 38734 LLinesclln 38964 LPlanesclpl 38965 |
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 5299 ax-nul 5306 ax-pr 5429 |
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 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-lplanes 38972 |
This theorem is referenced by: islpln4 39004 lplni 39005 lplnbase 39007 lplnnle2at 39014 |
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