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Theorem cdleme35h 35658
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one outside of  P  .\/  Q line. TODO: FIX COMMENT (Contributed by NM, 11-Mar-2013.)
Hypotheses
Ref Expression
cdleme35.l  |-  .<_  =  ( le `  K )
cdleme35.j  |-  .\/  =  ( join `  K )
cdleme35.m  |-  ./\  =  ( meet `  K )
cdleme35.a  |-  A  =  ( Atoms `  K )
cdleme35.h  |-  H  =  ( LHyp `  K
)
cdleme35.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme35.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
cdleme35.g  |-  G  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme35h  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  R  =  S )

Proof of Theorem cdleme35h
StepHypRef Expression
1 oveq1 6302 . . . . 5  |-  ( F  =  G  ->  ( F  .\/  U )  =  ( G  .\/  U
) )
2 oveq2 6303 . . . . . . 7  |-  ( F  =  G  ->  ( Q  .\/  F )  =  ( Q  .\/  G
) )
32oveq1d 6310 . . . . . 6  |-  ( F  =  G  ->  (
( Q  .\/  F
)  ./\  W )  =  ( ( Q 
.\/  G )  ./\  W ) )
43oveq2d 6311 . . . . 5  |-  ( F  =  G  ->  ( P  .\/  ( ( Q 
.\/  F )  ./\  W ) )  =  ( P  .\/  ( ( Q  .\/  G ) 
./\  W ) ) )
51, 4oveq12d 6313 . . . 4  |-  ( F  =  G  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
653ad2ant3 1019 . . 3  |-  ( ( -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  ->  ( ( F  .\/  U )  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
763ad2ant3 1019 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
8 simp1 996 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
9 simp21 1029 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  P  =/=  Q )
10 simp22 1030 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
11 simp31 1032 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
12 cdleme35.l . . . 4  |-  .<_  =  ( le `  K )
13 cdleme35.j . . . 4  |-  .\/  =  ( join `  K )
14 cdleme35.m . . . 4  |-  ./\  =  ( meet `  K )
15 cdleme35.a . . . 4  |-  A  =  ( Atoms `  K )
16 cdleme35.h . . . 4  |-  H  =  ( LHyp `  K
)
17 cdleme35.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme35.f . . . 4  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
1912, 13, 14, 15, 16, 17, 18cdleme35g 35657 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  R )
208, 9, 10, 11, 19syl121anc 1233 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  R )
21 simp23 1031 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
22 simp32 1033 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
23 cdleme35.g . . . 4  |-  G  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
2412, 13, 14, 15, 16, 17, 23cdleme35g 35657 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) )  =  S )
258, 9, 21, 22, 24syl121anc 1233 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  (
( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) )  =  S )
267, 20, 253eqtr3d 2516 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  R  =  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14578   joincjn 15447   meetcmee 15448   Atomscatm 34466   HLchlt 34553   LHypclh 35186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-poset 15449  df-plt 15461  df-lub 15477  df-glb 15478  df-join 15479  df-meet 15480  df-p0 15542  df-p1 15543  df-lat 15549  df-clat 15611  df-oposet 34379  df-ol 34381  df-oml 34382  df-covers 34469  df-ats 34470  df-atl 34501  df-cvlat 34525  df-hlat 34554  df-lines 34703  df-psubsp 34705  df-pmap 34706  df-padd 34998  df-lhyp 35190
This theorem is referenced by:  cdleme35h2  35659  cdleme36m  35663
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