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Theorem 2lplnm2N 33573
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 1022 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  Y  e.  P )
2 simp1 988 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  HL )
3 hllat 33316 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  Lat )
5 simp21 1021 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  e.  P )
6 eqid 2451 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2lplnm2.p . . . . . 6  |-  P  =  ( LPlanes `  K )
86, 7lplnbase 33486 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
95, 8syl 16 . . . 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 33486 . . . . 5  |-  ( Y  e.  P  ->  Y  e.  ( Base `  K
) )
111, 10syl 16 . . . 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 15326 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
144, 9, 11, 13syl3anc 1219 . . 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 2451 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
17 2lplnm2.v . . . . . . 7  |-  V  =  ( LVols `  K )
1815, 16, 7, 172lplnj 33572 . . . . . 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 1023 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  W  e.  V )
2018, 19eqeltrd 2539 . . . . 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 15334 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  X  .<_  ( X ( join `  K ) Y ) )
224, 9, 11, 21syl3anc 1219 . . . . 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 2451 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
2415, 23, 7, 17lplncvrlvol2 33567 . . . . 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 1222 . . . 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 33372 . . . . 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 1219 . . . 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 33510 . . 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 1222 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   meetcmee 15219   Latclat 15319    <o ccvr 33215   HLchlt 33303   LLinesclln 33443   LPlanesclpl 33444   LVolsclvol 33445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452
This theorem is referenced by: (None)
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