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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvoli | Structured version Visualization version GIF version |
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
lvolset.b | β’ π΅ = (BaseβπΎ) |
lvolset.c | β’ πΆ = ( β βπΎ) |
lvolset.p | β’ π = (LPlanesβπΎ) |
lvolset.v | β’ π = (LVolsβπΎ) |
Ref | Expression |
---|---|
lvoli | β’ (((πΎ β π· β§ π β π΅ β§ π β π) β§ ππΆπ) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 1192 | . 2 β’ (((πΎ β π· β§ π β π΅ β§ π β π) β§ ππΆπ) β π β π΅) | |
2 | breq1 5141 | . . . 4 β’ (π₯ = π β (π₯πΆπ β ππΆπ)) | |
3 | 2 | rspcev 3606 | . . 3 β’ ((π β π β§ ππΆπ) β βπ₯ β π π₯πΆπ) |
4 | 3 | 3ad2antl3 1187 | . 2 β’ (((πΎ β π· β§ π β π΅ β§ π β π) β§ ππΆπ) β βπ₯ β π π₯πΆπ) |
5 | simpl1 1191 | . . 3 β’ (((πΎ β π· β§ π β π΅ β§ π β π) β§ ππΆπ) β πΎ β π·) | |
6 | lvolset.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
7 | lvolset.c | . . . 4 β’ πΆ = ( β βπΎ) | |
8 | lvolset.p | . . . 4 β’ π = (LPlanesβπΎ) | |
9 | lvolset.v | . . . 4 β’ π = (LVolsβπΎ) | |
10 | 6, 7, 8, 9 | islvol 38233 | . . 3 β’ (πΎ β π· β (π β π β (π β π΅ β§ βπ₯ β π π₯πΆπ))) |
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 π₯πΆπ))) |
12 | 1, 4, 11 | mpbir2and 711 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwrex 3069 class class class wbr 5138 βcfv 6529 Basecbs 17123 β ccvr 37921 LPlanesclpl 38152 LVolsclvol 38153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6481 df-fun 6531 df-fv 6537 df-lvols 38160 |
This theorem is referenced by: lplncvrlvol 38276 |
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