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Theorem arglem1N 31001
Description: Lemma for Desargues' law. Theorem 13.3 of [Crawley] p. 110, 3rd and 4th lines from bottom. In these lemmas,  P,  Q,  R,  S,  T,  U,  C,  D,  E,  F, and  G represent Crawley's a0, a1, a2, b0, b1, b2, c, z0, z1, z2, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
arglem1.j  |-  .\/  =  ( join `  K )
arglem1.m  |-  ./\  =  ( meet `  K )
arglem1.a  |-  A  =  ( Atoms `  K )
arglem1.f  |-  F  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
arglem1.g  |-  G  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )
Assertion
Ref Expression
arglem1N  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  F  e.  A )

Proof of Theorem arglem1N
StepHypRef Expression
1 arglem1.f . 2  |-  F  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
2 simpl11 1030 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  K  e.  HL )
3 hllat 30175 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  K  e.  Lat )
5 simpl12 1031 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  P  e.  A )
6 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 arglem1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 30101 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
95, 8syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  P  e.  ( Base `  K
) )
10 simpl13 1032 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  Q  e.  A )
116, 7atbase 30101 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 11syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  Q  e.  ( Base `  K
) )
13 simpl21 1033 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  S  e.  A )
146, 7atbase 30101 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1513, 14syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  S  e.  ( Base `  K
) )
16 simpl22 1034 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  T  e.  A )
176, 7atbase 30101 . . . . . 6  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1816, 17syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  T  e.  ( Base `  K
) )
19 arglem1.j . . . . . 6  |-  .\/  =  ( join `  K )
206, 19latj4 14223 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K ) )  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( S  .\/  T ) )  =  ( ( P  .\/  S ) 
.\/  ( Q  .\/  T ) ) )
214, 9, 12, 15, 18, 20syl122anc 1191 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  (
( P  .\/  Q
)  .\/  ( S  .\/  T ) )  =  ( ( P  .\/  S )  .\/  ( Q 
.\/  T ) ) )
22 arglem1.g . . . . . 6  |-  G  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )
23 simpr 447 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  G  e.  A )
2422, 23syl5eqelr 2381 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  e.  A )
25 simpl31 1036 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  P  =/=  S )
26 eqid 2296 . . . . . . . 8  |-  ( LLines `  K )  =  (
LLines `  K )
2719, 7, 26llni2 30323 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  /\  P  =/=  S
)  ->  ( P  .\/  S )  e.  (
LLines `  K ) )
282, 5, 13, 25, 27syl31anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  ( P  .\/  S )  e.  ( LLines `  K )
)
29 simpl32 1037 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  Q  =/=  T )
3019, 7, 26llni2 30323 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
312, 10, 16, 29, 30syl31anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  ( Q  .\/  T )  e.  ( LLines `  K )
)
32 arglem1.m . . . . . . 7  |-  ./\  =  ( meet `  K )
33 eqid 2296 . . . . . . 7  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3419, 32, 7, 26, 332llnmj 30371 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  .\/  S )  e.  ( LLines `  K
)  /\  ( Q  .\/  T )  e.  (
LLines `  K ) )  ->  ( ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  e.  A  <->  ( ( P  .\/  S
)  .\/  ( Q  .\/  T ) )  e.  ( LPlanes `  K )
) )
352, 28, 31, 34syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  A  <->  ( ( P 
.\/  S )  .\/  ( Q  .\/  T ) )  e.  ( LPlanes `  K ) ) )
3624, 35mpbid 201 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  (
( P  .\/  S
)  .\/  ( Q  .\/  T ) )  e.  ( LPlanes `  K )
)
3721, 36eqeltrd 2370 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  (
( P  .\/  Q
)  .\/  ( S  .\/  T ) )  e.  ( LPlanes `  K )
)
38 simpl23 1035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  P  =/=  Q )
3919, 7, 26llni2 30323 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
402, 5, 10, 38, 39syl31anc 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  ( P  .\/  Q )  e.  ( LLines `  K )
)
41 simpl33 1038 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  S  =/=  T )
4219, 7, 26llni2 30323 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
432, 13, 16, 41, 42syl31anc 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  ( S  .\/  T )  e.  ( LLines `  K )
)
4419, 32, 7, 26, 332llnmj 30371 . . . 4  |-  ( ( K  e.  HL  /\  ( P  .\/  Q )  e.  ( LLines `  K
)  /\  ( S  .\/  T )  e.  (
LLines `  K ) )  ->  ( ( ( P  .\/  Q ) 
./\  ( S  .\/  T ) )  e.  A  <->  ( ( P  .\/  Q
)  .\/  ( S  .\/  T ) )  e.  ( LPlanes `  K )
) )
452, 40, 43, 44syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  (
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  e.  A  <->  ( ( P 
.\/  Q )  .\/  ( S  .\/  T ) )  e.  ( LPlanes `  K ) ) )
4637, 45mpbird 223 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  e.  A )
471, 46syl5eqel 2380 1  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  P  =/=  Q )  /\  ( P  =/=  S  /\  Q  =/=  T  /\  S  =/=  T
) )  /\  G  e.  A )  ->  F  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162   LLinesclln 30302   LPlanesclpl 30303
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-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
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