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Theorem 2lplnm2N 30432
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 989 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  Y  e.  P )
2 simp1 955 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  HL )
3 hllat 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  Lat )
5 simp21 988 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  e.  P )
6 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2lplnm2.p . . . . . 6  |-  P  =  ( LPlanes `  K )
86, 7lplnbase 30345 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
95, 8syl 15 . . . 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 30345 . . . . 5  |-  ( Y  e.  P  ->  Y  e.  ( Base `  K
) )
111, 10syl 15 . . . 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 14173 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
144, 9, 11, 13syl3anc 1182 . . 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 2296 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
17 2lplnm2.v . . . . . . 7  |-  V  =  ( LVols `  K )
1815, 16, 7, 172lplnj 30431 . . . . . 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 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  W  e.  V )
2018, 19eqeltrd 2370 . . . . 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 14182 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  X  .<_  ( X ( join `  K ) Y ) )
224, 9, 11, 21syl3anc 1182 . . . . 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 2296 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
2415, 23, 7, 17lplncvrlvol2 30426 . . . . 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 1185 . . . 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 30231 . . . . 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 1182 . . . 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 223 . . 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 30369 . . 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 1185 . 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 223 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167    <o ccvr 30074   HLchlt 30162   LLinesclln 30302   LPlanesclpl 30303   LVolsclvol 30304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311
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