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Theorem 2lplnm2N 32619
Description: The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2lplnm2.l  |-  .<_  =  ( le `  K )
2lplnm2.m  |-  ./\  =  ( meet `  K )
2lplnm2.a  |-  N  =  ( LLines `  K )
2lplnm2.p  |-  P  =  ( LPlanes `  K )
2lplnm2.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
2lplnm2N  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y )  e.  N )

Proof of Theorem 2lplnm2N
StepHypRef Expression
1 simp22 1031 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  Y  e.  P )
2 simp1 997 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  HL )
3 hllat 32362 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1018 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  Lat )
5 simp21 1030 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  e.  P )
6 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2lplnm2.p . . . . . 6  |-  P  =  ( LPlanes `  K )
86, 7lplnbase 32532 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
95, 8syl 17 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  e.  ( Base `  K
) )
106, 7lplnbase 32532 . . . . 5  |-  ( Y  e.  P  ->  Y  e.  ( Base `  K
) )
111, 10syl 17 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  Y  e.  ( Base `  K
) )
12 2lplnm2.m . . . . 5  |-  ./\  =  ( meet `  K )
136, 12latmcl 15898 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
144, 9, 11, 13syl3anc 1230 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
15 2lplnm2.l . . . . . . 7  |-  .<_  =  ( le `  K )
16 eqid 2402 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
17 2lplnm2.v . . . . . . 7  |-  V  =  ( LVols `  K )
1815, 16, 7, 172lplnj 32618 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X ( join `  K
) Y )  =  W )
19 simp23 1032 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  W  e.  V )
2018, 19eqeltrd 2490 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X ( join `  K
) Y )  e.  V )
216, 15, 16latlej1 15906 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  X  .<_  ( X ( join `  K ) Y ) )
224, 9, 11, 21syl3anc 1230 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  .<_  ( X ( join `  K ) Y ) )
23 eqid 2402 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
2415, 23, 7, 17lplncvrlvol2 32613 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  ( X ( join `  K
) Y )  e.  V )  /\  X  .<_  ( X ( join `  K ) Y ) )  ->  X (  <o  `  K ) ( X ( join `  K
) Y ) )
252, 5, 20, 22, 24syl31anc 1233 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X
(  <o  `  K )
( X ( join `  K ) Y ) )
266, 16, 12, 23cvrexch 32418 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  (
( X  ./\  Y
) (  <o  `  K
) Y  <->  X (  <o  `  K ) ( X ( join `  K
) Y ) ) )
272, 9, 11, 26syl3anc 1230 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( X  ./\  Y
) (  <o  `  K
) Y  <->  X (  <o  `  K ) ( X ( join `  K
) Y ) ) )
2825, 27mpbird 232 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y ) ( 
<o  `  K ) Y )
29 2lplnm2.a . . . 4  |-  N  =  ( LLines `  K )
306, 23, 29, 7llncvrlpln 32556 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  ./\  Y
)  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  /\  ( X  ./\  Y ) ( 
<o  `  K ) Y )  ->  ( ( X  ./\  Y )  e.  N  <->  Y  e.  P
) )
312, 14, 11, 28, 30syl31anc 1233 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( X  ./\  Y
)  e.  N  <->  Y  e.  P ) )
321, 31mpbird 232 1  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y )  e.  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   meetcmee 15790   Latclat 15891    <o ccvr 32261   HLchlt 32349   LLinesclln 32489   LPlanesclpl 32490   LVolsclvol 32491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-llines 32496  df-lplanes 32497  df-lvols 32498
This theorem is referenced by: (None)
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