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Theorem arglem1N 30826
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 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 )  ->  K  e.  HL )
3 hllat 30000 . . . . . 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 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 )  ->  P  e.  A )
6 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 arglem1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 29926 . . . . . 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 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 )  ->  Q  e.  A )
116, 7atbase 29926 . . . . . 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 1035 . . . . . 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 29926 . . . . . 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 1036 . . . . . 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 29926 . . . . . 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 14518 . . . . 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 1193 . . . 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 448 . . . . . 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 2520 . . . . 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 1038 . . . . . . 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 2435 . . . . . . . 8  |-  ( LLines `  K )  =  (
LLines `  K )
2719, 7, 26llni2 30148 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  /\  P  =/=  S
)  ->  ( P  .\/  S )  e.  (
LLines `  K ) )
282, 5, 13, 25, 27syl31anc 1187 . . . . . 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 1039 . . . . . . 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 30148 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
312, 10, 16, 29, 30syl31anc 1187 . . . . . 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 2435 . . . . . . 7  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3419, 32, 7, 26, 332llnmj 30196 . . . . . 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 1184 . . . . 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 202 . . . 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 2509 . . 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 1037 . . . . 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 30148 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
402, 5, 10, 38, 39syl31anc 1187 . . . 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 1040 . . . . 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 30148 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
432, 13, 16, 41, 42syl31anc 1187 . . . 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 30196 . . . 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 1184 . . 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 224 . 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 2519 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5445  (class class class)co 6072   Basecbs 13457   joincjn 14389   meetcmee 14390   Latclat 14462   Atomscatm 29900   HLchlt 29987   LLinesclln 30127   LPlanesclpl 30128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135
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