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Theorem cdleme21at 30590
Description: Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
Hypotheses
Ref Expression
cdleme21.l  |-  .<_  =  ( le `  K )
cdleme21.j  |-  .\/  =  ( join `  K )
cdleme21.m  |-  ./\  =  ( meet `  K )
cdleme21.a  |-  A  =  ( Atoms `  K )
cdleme21.h  |-  H  =  ( LHyp `  K
)
cdleme21.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme21at  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  U  .<_  ( S  .\/  T
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  T  =/=  z
)

Proof of Theorem cdleme21at
StepHypRef Expression
1 cdleme21.l . . . 4  |-  .<_  =  ( le `  K )
2 cdleme21.j . . . 4  |-  .\/  =  ( join `  K )
3 cdleme21.m . . . 4  |-  ./\  =  ( meet `  K )
4 cdleme21.a . . . 4  |-  A  =  ( Atoms `  K )
5 cdleme21.h . . . 4  |-  H  =  ( LHyp `  K
)
6 cdleme21.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
71, 2, 3, 4, 5, 6cdleme21c 30589 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  -.  U  .<_  ( S  .\/  z ) )
873adant2r 1177 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  U  .<_  ( S  .\/  T
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  -.  U  .<_  ( S  .\/  z ) )
9 simp2r 982 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  U  .<_  ( S  .\/  T
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  U  .<_  ( S 
.\/  T ) )
10 oveq2 5868 . . . . 5  |-  ( T  =  z  ->  ( S  .\/  T )  =  ( S  .\/  z
) )
1110breq2d 4037 . . . 4  |-  ( T  =  z  ->  ( U  .<_  ( S  .\/  T )  <->  U  .<_  ( S 
.\/  z ) ) )
129, 11syl5ibcom 211 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  U  .<_  ( S  .\/  T
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( T  =  z  ->  U  .<_  ( S  .\/  z ) ) )
1312necon3bd 2485 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  U  .<_  ( S  .\/  T
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( -.  U  .<_  ( S  .\/  z
)  ->  T  =/=  z ) )
148, 13mpd 14 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  U  .<_  ( S  .\/  T
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  T  =/=  z
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   lecple 13217   joincjn 14080   meetcmee 14081   Atomscatm 29526   HLchlt 29613   LHypclh 30246
This theorem is referenced by:  cdleme21e  30593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-lhyp 30250
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