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Theorem cdleme22a 30505
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t 
\/ v = p  \/ q implies v = u. (Contributed by NM, 30-Nov-2012.)
Hypotheses
Ref Expression
cdleme22.l  |-  .<_  =  ( le `  K )
cdleme22.j  |-  .\/  =  ( join `  K )
cdleme22.m  |-  ./\  =  ( meet `  K )
cdleme22.a  |-  A  =  ( Atoms `  K )
cdleme22.h  |-  H  =  ( LHyp `  K
)
cdleme22.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme22a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  V  =  U )

Proof of Theorem cdleme22a
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp21 990 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp22 991 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  Q  e.  A )
4 simp32 994 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  P  =/=  Q )
5 simp31l 1080 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  V  e.  A )
6 simp31r 1081 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  V  .<_  W )
7 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  K  e.  HL )
8 simp23 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  T  e.  A )
9 cdleme22.l . . . . 5  |-  .<_  =  ( le `  K )
10 cdleme22.j . . . . 5  |-  .\/  =  ( join `  K )
11 cdleme22.a . . . . 5  |-  A  =  ( Atoms `  K )
129, 10, 11hlatlej2 29541 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  V  .<_  ( T  .\/  V ) )
137, 8, 5, 12syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  V  .<_  ( T  .\/  V
) )
14 simp33 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  ( T  .\/  V )  =  ( P  .\/  Q
) )
1513, 14breqtrd 4170 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  V  .<_  ( P  .\/  Q
) )
16 cdleme22.m . . 3  |-  ./\  =  ( meet `  K )
17 cdleme22.h . . 3  |-  H  =  ( LHyp `  K
)
18 cdleme22.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
199, 10, 16, 11, 17, 18cdleme22aa 30504 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( P  .\/  Q ) ) )  ->  V  =  U )
201, 2, 3, 4, 5, 6, 15, 19syl133anc 1207 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  V  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   lecple 13456   joincjn 14321   meetcmee 14322   Atomscatm 29429   HLchlt 29516   LHypclh 30149
This theorem is referenced by:  cdleme22e  30509  cdleme22eALTN  30510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153
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