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Theorem cdleme42ke 36354
Description: Part of proof of Lemma E in [Crawley] p. 113. Remove  R  =/= 
S condition. TODO: FIX COMMENT (Contributed by NM, 2-Apr-2013.)
Hypotheses
Ref Expression
cdleme41.b  |-  B  =  ( Base `  K
)
cdleme41.l  |-  .<_  =  ( le `  K )
cdleme41.j  |-  .\/  =  ( join `  K )
cdleme41.m  |-  ./\  =  ( meet `  K )
cdleme41.a  |-  A  =  ( Atoms `  K )
cdleme41.h  |-  H  =  ( LHyp `  K
)
cdleme41.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme41.d  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme41.e  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme41.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme41.i  |-  I  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
cdleme41.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme41.o  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme41.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
cdleme34e.v  |-  V  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme42ke  |-  ( ( ( ( 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 ) ) )  ->  ( ( F `
 R )  .\/  ( F `  S ) )  =  ( ( F `  R ) 
.\/  V ) )
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, s    S, s    U, s    W, s    y, t, A, s    B, s, t, y    y, D    y, G    E, s,
y    H, s, t, y   
t,  .\/ , y    K, s, t, y    t,  .<_ , y   
t,  ./\ , y    t, P, y    t, Q, y    t, R, y    t, S, y   
t, U, y    t, W, y    x, z, A   
x, B, z    z, E, s    z, H    x,  .\/ , z    z, K    x,  .<_ , z    x,  ./\ , z    x, N, z    x, P, z   
x, Q, z    x, R, z    x, S, z   
x, U, z    x, W, z, s, t, y    V, s, t, x, z
Allowed substitution hints:    D( x, z, t, s)    E( x, t)    F( x, y, z, t, s)    G( x, z, t, s)    H( x)    I( x, y, z, t, s)    K( x)    N( y, t, s)    O( x, y, z, t, s)    V( y)

Proof of Theorem cdleme42ke
StepHypRef Expression
1 simpl1l 1047 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  K  e.  HL )
2 simpr2 1003 . . . . . . 7  |-  ( ( ( ( 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  e.  A  /\  -.  R  .<_  W ) )
3 cdleme41.b . . . . . . . 8  |-  B  =  ( Base `  K
)
4 cdleme41.l . . . . . . . 8  |-  .<_  =  ( le `  K )
5 cdleme41.j . . . . . . . 8  |-  .\/  =  ( join `  K )
6 cdleme41.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
7 cdleme41.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
8 cdleme41.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
9 cdleme41.u . . . . . . . 8  |-  U  =  ( ( P  .\/  Q )  ./\  W )
10 cdleme41.d . . . . . . . 8  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
11 cdleme41.e . . . . . . . 8  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
12 cdleme41.g . . . . . . . 8  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
13 cdleme41.i . . . . . . . 8  |-  I  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
14 cdleme41.n . . . . . . . 8  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
15 cdleme41.o . . . . . . . 8  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
16 cdleme41.f . . . . . . . 8  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
173, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16cdleme32fvaw 36308 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
182, 17syldan 470 . . . . . 6  |-  ( ( ( ( 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 ) ) )  ->  ( ( F `
 R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
1918simpld 459 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  ( F `  R )  e.  A
)
205, 7hlatjidm 35236 . . . . 5  |-  ( ( K  e.  HL  /\  ( F `  R )  e.  A )  -> 
( ( F `  R )  .\/  ( F `  R )
)  =  ( F `
 R ) )
211, 19, 20syl2anc 661 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  ( ( F `
 R )  .\/  ( F `  R ) )  =  ( F `
 R ) )
22 fveq2 5872 . . . . 5  |-  ( R  =  S  ->  ( F `  R )  =  ( F `  S ) )
2322oveq2d 6312 . . . 4  |-  ( R  =  S  ->  (
( F `  R
)  .\/  ( F `  R ) )  =  ( ( F `  R )  .\/  ( F `  S )
) )
2421, 23sylan9req 2519 . . 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  =  S )  ->  ( F `  R )  =  ( ( F `  R
)  .\/  ( F `  S ) ) )
25 simpr2l 1055 . . . . . . . . 9  |-  ( ( ( ( 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  e.  A
)
265, 7hlatjidm 35236 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
271, 25, 26syl2anc 661 . . . . . . . 8  |-  ( ( ( ( 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  .\/  R )  =  R )
2827oveq1d 6311 . . . . . . 7  |-  ( ( ( ( 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 
.\/  R )  ./\  W )  =  ( R 
./\  W ) )
29 simpl1 999 . . . . . . . 8  |-  ( ( ( ( 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 ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
30 eqid 2457 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
314, 6, 30, 7, 8lhpmat 35897 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
3229, 2, 31syl2anc 661 . . . . . . 7  |-  ( ( ( ( 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  ./\  W )  =  ( 0.
`  K ) )
3328, 32eqtrd 2498 . . . . . 6  |-  ( ( ( ( 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 
.\/  R )  ./\  W )  =  ( 0.
`  K ) )
3433oveq2d 6312 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  ( ( F `
 R )  .\/  ( ( R  .\/  R )  ./\  W )
)  =  ( ( F `  R ) 
.\/  ( 0. `  K ) ) )
35 hlol 35229 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
361, 35syl 16 . . . . . 6  |-  ( ( ( ( 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 ) ) )  ->  K  e.  OL )
373, 7atbase 35157 . . . . . . 7  |-  ( ( F `  R )  e.  A  ->  ( F `  R )  e.  B )
3819, 37syl 16 . . . . . 6  |-  ( ( ( ( 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 ) ) )  ->  ( F `  R )  e.  B
)
393, 5, 30olj01 35093 . . . . . 6  |-  ( ( K  e.  OL  /\  ( F `  R )  e.  B )  -> 
( ( F `  R )  .\/  ( 0. `  K ) )  =  ( F `  R ) )
4036, 38, 39syl2anc 661 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  ( ( F `
 R )  .\/  ( 0. `  K ) )  =  ( F `
 R ) )
4134, 40eqtrd 2498 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  ( ( F `
 R )  .\/  ( ( R  .\/  R )  ./\  W )
)  =  ( F `
 R ) )
42 oveq2 6304 . . . . . . 7  |-  ( R  =  S  ->  ( R  .\/  R )  =  ( R  .\/  S
) )
4342oveq1d 6311 . . . . . 6  |-  ( R  =  S  ->  (
( R  .\/  R
)  ./\  W )  =  ( ( R 
.\/  S )  ./\  W ) )
44 cdleme34e.v . . . . . 6  |-  V  =  ( ( R  .\/  S )  ./\  W )
4543, 44syl6eqr 2516 . . . . 5  |-  ( R  =  S  ->  (
( R  .\/  R
)  ./\  W )  =  V )
4645oveq2d 6312 . . . 4  |-  ( R  =  S  ->  (
( F `  R
)  .\/  ( ( R  .\/  R )  ./\  W ) )  =  ( ( F `  R
)  .\/  V )
)
4741, 46sylan9req 2519 . . 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  =  S )  ->  ( F `  R )  =  ( ( F `  R
)  .\/  V )
)
4824, 47eqtr3d 2500 . 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  =  S )  ->  ( ( F `  R )  .\/  ( F `  S
) )  =  ( ( F `  R
)  .\/  V )
)
493, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44cdleme42k 36353 . . 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  =/=  S )  -> 
( ( F `  R )  .\/  ( F `  S )
)  =  ( ( F `  R ) 
.\/  V ) )
50493expa 1196 . 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  =/=  S
)  ->  ( ( F `  R )  .\/  ( F `  S
) )  =  ( ( F `  R
)  .\/  V )
)
5148, 50pm2.61dane 2775 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 ) ) )  ->  ( ( F `
 R )  .\/  ( F `  S ) )  =  ( ( F `  R ) 
.\/  V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   meetcmee 15701   0.cp0 15794   OLcol 35042   Atomscatm 35131   HLchlt 35218   LHypclh 35851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855
This theorem is referenced by:  cdleme42keg  36355  cdleme42mN  36356  cdlemeg46fjv  36392
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