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Theorem arglem1N 33726
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 1080 . . . . . 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 32899 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . . . 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 1081 . . . . . 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 2422 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 arglem1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 32825 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
95, 8syl 17 . . . . 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 1082 . . . . . 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 32825 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 11syl 17 . . . . 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 1083 . . . . . 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 32825 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1513, 14syl 17 . . . . 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 1084 . . . . . 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 32825 . . . . . 6  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1816, 17syl 17 . . . . 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 16347 . . . . 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 1273 . . . 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 462 . . . . . 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 2512 . . . . 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 1086 . . . . . . 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 2422 . . . . . . . 8  |-  ( LLines `  K )  =  (
LLines `  K )
2719, 7, 26llni2 33047 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  /\  P  =/=  S
)  ->  ( P  .\/  S )  e.  (
LLines `  K ) )
282, 5, 13, 25, 27syl31anc 1267 . . . . . 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 1087 . . . . . . 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 33047 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
312, 10, 16, 29, 30syl31anc 1267 . . . . . 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 2422 . . . . . . 7  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3419, 32, 7, 26, 332llnmj 33095 . . . . . 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 1264 . . . . 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 213 . . . 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 2507 . . 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 1085 . . . . 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 33047 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
402, 5, 10, 38, 39syl31anc 1267 . . . 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 1088 . . . . 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 33047 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
432, 13, 16, 41, 42syl31anc 1267 . . . 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 33095 . . . 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 1264 . . 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 235 . 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 2511 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   ` cfv 5601  (class class class)co 6306   Basecbs 15121   joincjn 16189   meetcmee 16190   Latclat 16291   Atomscatm 32799   HLchlt 32886   LLinesclln 33026   LPlanesclpl 33027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-lat 16292  df-clat 16354  df-oposet 32712  df-ol 32714  df-oml 32715  df-covers 32802  df-ats 32803  df-atl 32834  df-cvlat 32858  df-hlat 32887  df-llines 33033  df-lplanes 33034
This theorem is referenced by: (None)
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