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Theorem cdleme22a 34293
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 988 . 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 1021 . 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 1022 . 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 1025 . 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 1111 . 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 1112 . 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 1012 . . . 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 1023 . . . 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 33329 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  V  .<_  ( T  .\/  V ) )
137, 8, 5, 12syl3anc 1219 . . 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 1026 . . 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 4417 . 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 34292 . 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 1242 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   lecple 14356   joincjn 15225   meetcmee 15226   Atomscatm 33217   HLchlt 33304   LHypclh 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-lhyp 33941
This theorem is referenced by:  cdleme22e  34297  cdleme22eALTN  34298
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