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Theorem cdleme39a 34103
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on  P  .\/  Q line. TODO: FIX COMMENT.  E,  Y,  G,  Z serve as f(t), f(u), ft( R), ft( S). Put hypotheses of cdleme38n 34102 in convention of cdleme32sn1awN 34070. TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013.)
Hypotheses
Ref Expression
cdleme39.l  |-  .<_  =  ( le `  K )
cdleme39.j  |-  .\/  =  ( join `  K )
cdleme39.m  |-  ./\  =  ( meet `  K )
cdleme39.a  |-  A  =  ( Atoms `  K )
cdleme39.h  |-  H  =  ( LHyp `  K
)
cdleme39.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme39.e  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme39.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )
cdleme39a.v  |-  V  =  ( ( t  .\/  E )  ./\  W )
Assertion
Ref Expression
cdleme39a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  G  =  ( ( R  .\/  V
)  ./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) ) )

Proof of Theorem cdleme39a
StepHypRef Expression
1 cdleme39.g . 2  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )
2 simp11 1060 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp12 1061 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  P  e.  A
)
4 simp13 1062 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  Q  e.  A
)
5 simp2 1031 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
6 simp3l 1058 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  R  .<_  ( P 
.\/  Q ) )
7 cdleme39.l . . . . . 6  |-  .<_  =  ( le `  K )
8 cdleme39.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdleme39.m . . . . . 6  |-  ./\  =  ( meet `  K )
10 cdleme39.a . . . . . 6  |-  A  =  ( Atoms `  K )
11 cdleme39.h . . . . . 6  |-  H  =  ( LHyp `  K
)
12 cdleme39.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
137, 8, 9, 10, 11, 12cdleme4 33875 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P 
.\/  Q ) )  ->  ( P  .\/  Q )  =  ( R 
.\/  U ) )
142, 3, 4, 5, 6, 13syl131anc 1305 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( P  .\/  Q )  =  ( R 
.\/  U ) )
15 cdleme39a.v . . . . . 6  |-  V  =  ( ( t  .\/  E )  ./\  W )
16 simp3r 1059 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( t  e.  A  /\  -.  t  .<_  W ) )
17 cdleme39.e . . . . . . . 8  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
187, 8, 9, 10, 11, 12, 17cdleme2 33865 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( (
t  .\/  E )  ./\  W )  =  U )
192, 3, 4, 16, 18syl13anc 1294 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( ( t 
.\/  E )  ./\  W )  =  U )
2015, 19syl5eq 2517 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  V  =  U )
2120oveq2d 6324 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( R  .\/  V )  =  ( R 
.\/  U ) )
2214, 21eqtr4d 2508 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( P  .\/  Q )  =  ( R 
.\/  V ) )
23 simp11l 1141 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  K  e.  HL )
24 simp2l 1056 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  R  e.  A
)
25 simp3rl 1103 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  t  e.  A
)
268, 10hlatjcom 33004 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  t  e.  A )  ->  ( R  .\/  t
)  =  ( t 
.\/  R ) )
2723, 24, 25, 26syl3anc 1292 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( R  .\/  t )  =  ( t  .\/  R ) )
2827oveq1d 6323 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( ( R 
.\/  t )  ./\  W )  =  ( ( t  .\/  R ) 
./\  W ) )
2928oveq2d 6324 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( E  .\/  ( ( R  .\/  t )  ./\  W
) )  =  ( E  .\/  ( ( t  .\/  R ) 
./\  W ) ) )
3022, 29oveq12d 6326 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  ( ( P 
.\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t ) 
./\  W ) ) )  =  ( ( R  .\/  V ) 
./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) ) )
311, 30syl5eq 2517 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P  .\/  Q )  /\  ( t  e.  A  /\  -.  t  .<_  W ) ) )  ->  G  =  ( ( R  .\/  V
)  ./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   HLchlt 32987   LHypclh 33620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624
This theorem is referenced by:  cdleme39n  34104
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