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Theorem cdleme39n 33498
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 33496 in convention of cdleme32sn1awN 33464. 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
) ) )
cdleme39.y  |-  Y  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
cdleme39.z  |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( S  .\/  u )  ./\  W
) ) )
Assertion
Ref Expression
cdleme39n  |-  ( ( ( ( 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  G  =/=  Z )

Proof of Theorem cdleme39n
StepHypRef Expression
1 cdleme39.l . . 3  |-  .<_  =  ( le `  K )
2 cdleme39.j . . 3  |-  .\/  =  ( join `  K )
3 cdleme39.m . . 3  |-  ./\  =  ( meet `  K )
4 cdleme39.a . . 3  |-  A  =  ( Atoms `  K )
5 cdleme39.h . . 3  |-  H  =  ( LHyp `  K
)
6 cdleme39.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 cdleme39.e . . 3  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
8 cdleme39.y . . 3  |-  Y  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
9 eqid 2404 . . 3  |-  ( ( t  .\/  E ) 
./\  W )  =  ( ( t  .\/  E )  ./\  W )
10 eqid 2404 . . 3  |-  ( ( u  .\/  Y ) 
./\  W )  =  ( ( u  .\/  Y )  ./\  W )
11 eqid 2404 . . 3  |-  ( ( R  .\/  ( ( t  .\/  E ) 
./\  W ) ) 
./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) )  =  ( ( R  .\/  (
( t  .\/  E
)  ./\  W )
)  ./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) )
12 eqid 2404 . . 3  |-  ( ( S  .\/  ( ( u  .\/  Y ) 
./\  W ) ) 
./\  ( Y  .\/  ( ( u  .\/  S )  ./\  W )
) )  =  ( ( S  .\/  (
( u  .\/  Y
)  ./\  W )
)  ./\  ( Y  .\/  ( ( u  .\/  S )  ./\  W )
) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdleme38n 33496 . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( R  .\/  ( ( t  .\/  E )  ./\  W )
)  ./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) )  =/=  (
( S  .\/  (
( u  .\/  Y
)  ./\  W )
)  ./\  ( Y  .\/  ( ( u  .\/  S )  ./\  W )
) ) )
14 simp11 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
15 simp12l 1112 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  P  e.  A )
16 simp13l 1114 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  Q  e.  A )
17 simp22l 1118 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  R  e.  A )
18 simp22r 1119 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  -.  R  .<_  W )
19 simp311 1146 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  R  .<_  ( P  .\/  Q ) )
20 simp32l 1124 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( t  e.  A  /\  -.  t  .<_  W ) )
21 cdleme39.g . . . 4  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )
221, 2, 3, 4, 5, 6, 7, 21, 9cdleme39a 33497 . . 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 ) ) )  ->  G  =  ( ( R  .\/  (
( t  .\/  E
)  ./\  W )
)  ./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) ) )
2314, 15, 16, 17, 18, 19, 20, 22syl322anc 1260 . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  G  =  ( ( R  .\/  ( ( t 
.\/  E )  ./\  W ) )  ./\  ( E  .\/  ( ( t 
.\/  R )  ./\  W ) ) ) )
24 simp23l 1120 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  S  e.  A )
25 simp23r 1121 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  -.  S  .<_  W )
26 simp312 1147 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  S  .<_  ( P  .\/  Q ) )
27 simp33l 1126 . . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( u  e.  A  /\  -.  u  .<_  W ) )
28 cdleme39.z . . . 4  |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( S  .\/  u )  ./\  W
) ) )
291, 2, 3, 4, 5, 6, 8, 28, 10cdleme39a 33497 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .<_  ( P  .\/  Q )  /\  ( u  e.  A  /\  -.  u  .<_  W ) ) )  ->  Z  =  ( ( S  .\/  (
( u  .\/  Y
)  ./\  W )
)  ./\  ( Y  .\/  ( ( u  .\/  S )  ./\  W )
) ) )
3014, 15, 16, 24, 25, 26, 27, 29syl322anc 1260 . 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  Z  =  ( ( S  .\/  ( ( u 
.\/  Y )  ./\  W ) )  ./\  ( Y  .\/  ( ( u 
.\/  S )  ./\  W ) ) ) )
3113, 23, 303netr4d 2710 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 )  /\  R  =/=  S )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  G  =/=  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   lecple 14918   joincjn 15899   meetcmee 15900   Atomscatm 32294   HLchlt 32381   LHypclh 33014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528  df-lplanes 32529  df-lvols 32530  df-lines 32531  df-psubsp 32533  df-pmap 32534  df-padd 32826  df-lhyp 33018
This theorem is referenced by:  cdleme40m  33499
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