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Theorem cdleme20l1 29776
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l  |-  .<_  =  ( le `  K )
cdleme19.j  |-  .\/  =  ( join `  K )
cdleme19.m  |-  ./\  =  ( meet `  K )
cdleme19.a  |-  A  =  ( Atoms `  K )
cdleme19.h  |-  H  =  ( LHyp `  K
)
cdleme19.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme19.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme19.g  |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme19.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
cdleme19.y  |-  Y  =  ( ( R  .\/  T )  ./\  W )
cdleme20.v  |-  V  =  ( ( S  .\/  T )  ./\  W )
Assertion
Ref Expression
cdleme20l1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( F  .\/  D )  e.  ( LLines `  K )
)

Proof of Theorem cdleme20l1
StepHypRef Expression
1 simp11l 1068 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
2 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp12 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp13 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
5 simp22 991 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  S  e.  A )
6 simp23 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  W )
75, 6jca 520 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
8 simp31 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  Q )
9 simp32 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
10 cdleme19.l . . . 4  |-  .<_  =  ( le `  K )
11 cdleme19.j . . . 4  |-  .\/  =  ( join `  K )
12 cdleme19.m . . . 4  |-  ./\  =  ( meet `  K )
13 cdleme19.a . . . 4  |-  A  =  ( Atoms `  K )
14 cdleme19.h . . . 4  |-  H  =  ( LHyp `  K
)
15 cdleme19.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
16 cdleme19.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
1710, 11, 12, 13, 14, 15, 16cdleme3fa 29692 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  F  e.  A )
182, 3, 4, 7, 8, 9, 17syl132anc 1202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  F  e.  A )
19 simp11r 1069 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  H )
20 simp21 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
21 simp33 995 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q
) )
22 cdleme19.d . . . 4  |-  D  =  ( ( R  .\/  S )  ./\  W )
2310, 11, 12, 13, 14, 22cdlemeda 29754 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  D  e.  A )
241, 19, 5, 6, 20, 21, 9, 23syl223anc 1210 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  D  e.  A )
2510, 11, 12, 13, 14, 15, 16, 16, 22, 22cdleme19c 29761 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  F  =/=  D )
261, 19, 3, 4, 7, 20, 8, 9, 25syl233anc 1213 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  F  =/=  D )
27 eqid 2284 . . 3  |-  ( LLines `  K )  =  (
LLines `  K )
2811, 13, 27llni2 28968 . 2  |-  ( ( ( K  e.  HL  /\  F  e.  A  /\  D  e.  A )  /\  F  =/=  D
)  ->  ( F  .\/  D )  e.  (
LLines `  K ) )
291, 18, 24, 26, 28syl31anc 1187 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( F  .\/  D )  e.  ( LLines `  K )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LLinesclln 28947   LHypclh 29440
This theorem is referenced by:  cdleme20l2  29777  cdleme20l  29778
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  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 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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-iin 3909  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-p1 14140  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-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444
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