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Theorem cdleme20k 29638
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (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
cdleme20k  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( F  .\/  D )  =/=  ( P  .\/  Q ) )

Proof of Theorem cdleme20k
StepHypRef Expression
1 simp11 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
3 simp13 992 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
4 simp2r 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
5 simp2l 986 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
6 simp3r 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q ) )
7 simp3l 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q
) )
8 cdleme19.l . . . 4  |-  .<_  =  ( le `  K )
9 cdleme19.j . . . 4  |-  .\/  =  ( join `  K )
10 cdleme19.m . . . 4  |-  ./\  =  ( meet `  K )
11 cdleme19.a . . . 4  |-  A  =  ( Atoms `  K )
12 cdleme19.h . . . 4  |-  H  =  ( LHyp `  K
)
13 cdleme19.d . . . 4  |-  D  =  ( ( R  .\/  S )  ./\  W )
148, 9, 10, 11, 12, 13cdlemednpq 29618 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  D  .<_  ( P 
.\/  Q ) )
151, 2, 3, 4, 5, 6, 7, 14syl133anc 1210 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  -.  D  .<_  ( P  .\/  Q
) )
16 simp11l 1071 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
17 hllat 28683 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1816, 17syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
19 simp11r 1072 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  W  e.  H )
20 simp2ll 1027 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  A )
21 cdleme19.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
22 cdleme19.f . . . . . . 7  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
23 eqid 2256 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
248, 9, 10, 11, 12, 21, 22, 23cdleme1b 29545 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A ) )  ->  F  e.  ( Base `  K ) )
2516, 19, 2, 3, 20, 24syl23anc 1194 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  F  e.  ( Base `  K )
)
26 simp2rl 1029 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  R  e.  A )
278, 9, 10, 11, 12, 13, 23cdlemedb 29616 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  D  e.  ( Base `  K ) )
2816, 19, 26, 20, 27syl22anc 1188 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  D  e.  ( Base `  K )
)
2923, 8, 9latlej2 14094 . . . . 5  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  D  e.  ( Base `  K
) )  ->  D  .<_  ( F  .\/  D
) )
3018, 25, 28, 29syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  D  .<_  ( F  .\/  D ) )
31 breq2 3967 . . . 4  |-  ( ( F  .\/  D )  =  ( P  .\/  Q )  ->  ( D  .<_  ( F  .\/  D
)  <->  D  .<_  ( P 
.\/  Q ) ) )
3230, 31syl5ibcom 213 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( ( F  .\/  D )  =  ( P  .\/  Q
)  ->  D  .<_  ( P  .\/  Q ) ) )
3332necon3bd 2456 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( -.  D  .<_  ( P  .\/  Q )  ->  ( F  .\/  D )  =/=  ( P  .\/  Q ) ) )
3415, 33mpd 16 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( F  .\/  D )  =/=  ( P  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   Latclat 14078   Atomscatm 28583   HLchlt 28670   LHypclh 29303
This theorem is referenced by:  cdleme20l  29641
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307
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