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Theorem cdleme22a 29796
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 960 . 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 993 . 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 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 ) ) )  ->  Q  e.  A )
4 simp32 997 . 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 1083 . 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 1084 . 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 984 . . . 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 995 . . . 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 28832 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  V  .<_  ( T  .\/  V ) )
137, 8, 5, 12syl3anc 1187 . . 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 998 . . 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 4048 . 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 29795 . 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 1210 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440
This theorem is referenced by:  cdleme22e  29800  cdleme22eALTN  29801
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-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-lhyp 29444
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