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Theorem arglem1N 29646
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 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 )  ->  K  e.  HL )
3 hllat 28820 . . . . . 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 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 )  ->  P  e.  A )
6 eqid 2284 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 arglem1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 28746 . . . . . 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 1037 . . . . . 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 28746 . . . . . 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 1038 . . . . . 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 28746 . . . . . 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 1039 . . . . . 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 28746 . . . . . 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 14201 . . . . 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 1196 . . . 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 449 . . . . . 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 2369 . . . . 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 1041 . . . . . . 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 2284 . . . . . . . 8  |-  ( LLines `  K )  =  (
LLines `  K )
2719, 7, 26llni2 28968 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  /\  P  =/=  S
)  ->  ( P  .\/  S )  e.  (
LLines `  K ) )
282, 5, 13, 25, 27syl31anc 1190 . . . . . 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 1042 . . . . . . 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 28968 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
312, 10, 16, 29, 30syl31anc 1190 . . . . . 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 2284 . . . . . . 7  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3419, 32, 7, 26, 332llnmj 29016 . . . . . 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 1187 . . . . 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 203 . . . 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 2358 . . 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 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 )  ->  P  =/=  Q )
3919, 7, 26llni2 28968 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
402, 5, 10, 38, 39syl31anc 1190 . . . 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 1043 . . . . 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 28968 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
432, 13, 16, 41, 42syl31anc 1190 . . . 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 29016 . . . 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 1187 . . 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 225 . 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 2368 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   ` cfv 5221  (class class class)co 5819   Basecbs 13142   joincjn 14072   meetcmee 14073   Latclat 14145   Atomscatm 28720   HLchlt 28807   LLinesclln 28947   LPlanesclpl 28948
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955
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