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Theorem arglem1N 36016
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 1071 . . . . . 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 35189 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . . . 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 1072 . . . . . 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 2457 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 arglem1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 35115 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
95, 8syl 16 . . . . 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 1073 . . . . . 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 35115 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 11syl 16 . . . . 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 1074 . . . . . 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 35115 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1513, 14syl 16 . . . . 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 1075 . . . . . 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 35115 . . . . . 6  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1816, 17syl 16 . . . . 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 15857 . . . . 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 1237 . . . 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 461 . . . . . 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 2550 . . . . 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 1077 . . . . . . 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 2457 . . . . . . . 8  |-  ( LLines `  K )  =  (
LLines `  K )
2719, 7, 26llni2 35337 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  /\  P  =/=  S
)  ->  ( P  .\/  S )  e.  (
LLines `  K ) )
282, 5, 13, 25, 27syl31anc 1231 . . . . . 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 1078 . . . . . . 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 35337 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
312, 10, 16, 29, 30syl31anc 1231 . . . . . 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 2457 . . . . . . 7  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3419, 32, 7, 26, 332llnmj 35385 . . . . . 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 1228 . . . . 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 210 . . . 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 2545 . . 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 1076 . . . . 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 35337 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
402, 5, 10, 38, 39syl31anc 1231 . . . 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 1079 . . . . 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 35337 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
432, 13, 16, 41, 42syl31anc 1231 . . . 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 35385 . . . 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 1228 . . 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 232 . 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 2549 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   ` cfv 5594  (class class class)co 6296   Basecbs 14643   joincjn 15699   meetcmee 15700   Latclat 15801   Atomscatm 35089   HLchlt 35176   LLinesclln 35316   LPlanesclpl 35317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324
This theorem is referenced by: (None)
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