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Theorem cdleme35h 34112
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 6110 . . . . 5  |-  ( F  =  G  ->  ( F  .\/  U )  =  ( G  .\/  U
) )
2 oveq2 6111 . . . . . . 7  |-  ( F  =  G  ->  ( Q  .\/  F )  =  ( Q  .\/  G
) )
32oveq1d 6118 . . . . . 6  |-  ( F  =  G  ->  (
( Q  .\/  F
)  ./\  W )  =  ( ( Q 
.\/  G )  ./\  W ) )
43oveq2d 6119 . . . . 5  |-  ( F  =  G  ->  ( P  .\/  ( ( Q 
.\/  F )  ./\  W ) )  =  ( P  .\/  ( ( Q  .\/  G ) 
./\  W ) ) )
51, 4oveq12d 6121 . . . 4  |-  ( F  =  G  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
653ad2ant3 1011 . . 3  |-  ( ( -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  ->  ( ( F  .\/  U )  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
763ad2ant3 1011 . 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 988 . . 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 1021 . . 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 1022 . . 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 1024 . . 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 34111 . . 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 1223 . 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 1023 . . 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 1025 . . 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 34111 . . 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 1223 . 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 2483 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   lecple 14257   joincjn 15126   meetcmee 15127   Atomscatm 32920   HLchlt 33007   LHypclh 33640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-p1 15222  df-lat 15228  df-clat 15290  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-lines 33157  df-psubsp 33159  df-pmap 33160  df-padd 33452  df-lhyp 33644
This theorem is referenced by:  cdleme35h2  34113  cdleme36m  34117
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