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Theorem cdleme50trn3 34197
Description: Part of proof that  F is a translation.  P  =  Q case. TODO: fix comment. (Contributed by NM, 10-Apr-2013.)
Hypotheses
Ref Expression
cdlemef50.b  |-  B  =  ( Base `  K
)
cdlemef50.l  |-  .<_  =  ( le `  K )
cdlemef50.j  |-  .\/  =  ( join `  K )
cdlemef50.m  |-  ./\  =  ( meet `  K )
cdlemef50.a  |-  A  =  ( Atoms `  K )
cdlemef50.h  |-  H  =  ( LHyp `  K
)
cdlemef50.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef50.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs50.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef50.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
Assertion
Ref Expression
cdleme50trn3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  (
( R  .\/  ( F `  R )
)  ./\  W )  =  U )
Distinct variable groups:    t, s, x, y, z,  ./\    .\/ , s,
t, x, y, z    .<_ , s, t, x, y, z    A, s, t, x, y, z    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    K, s, t, x, y, z    P, s, t, x, y, z    Q, s, t, x, y, z    R, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z
Allowed substitution hints:    D( t)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme50trn3
StepHypRef Expression
1 simpl1 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simprr 756 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
3 cdlemef50.l . . . . . 6  |-  .<_  =  ( le `  K )
4 cdlemef50.m . . . . . 6  |-  ./\  =  ( meet `  K )
5 eqid 2443 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 cdlemef50.a . . . . . 6  |-  A  =  ( Atoms `  K )
7 cdlemef50.h . . . . . 6  |-  H  =  ( LHyp `  K
)
83, 4, 5, 6, 7lhpmat 33674 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
91, 2, 8syl2anc 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 ) ) )  ->  ( R  ./\  W )  =  ( 0. `  K
) )
10 simprrl 763 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  A )
11 cdlemef50.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1211, 6atbase 32934 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  B )
1310, 12syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  B )
14 simprl 755 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  =  Q )
15 cdlemef50.f . . . . . . . . 9  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
1615cdleme31id 34038 . . . . . . . 8  |-  ( ( R  e.  B  /\  P  =  Q )  ->  ( F `  R
)  =  R )
1713, 14, 16syl2anc 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 ) ) )  ->  ( F `  R )  =  R )
1817oveq2d 6107 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  ( F `  R ) )  =  ( R  .\/  R
) )
19 simpl1l 1039 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  HL )
20 cdlemef50.j . . . . . . . 8  |-  .\/  =  ( join `  K )
2120, 6hlatjidm 33013 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
2219, 10, 21syl2anc 661 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  R )  =  R )
2318, 22eqtrd 2475 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  ( F `  R ) )  =  R )
2423oveq1d 6106 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  (
( R  .\/  ( F `  R )
)  ./\  W )  =  ( R  ./\  W ) )
25 simpl2 992 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
263, 4, 5, 6, 7lhpmat 33674 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
271, 25, 26syl2anc 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 ) ) )  ->  ( P  ./\  W )  =  ( 0. `  K
) )
289, 24, 273eqtr4d 2485 . . 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  .\/  ( F `  R )
)  ./\  W )  =  ( P  ./\  W ) )
29 simpl2l 1041 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  A )
3020, 6hlatjidm 33013 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
3119, 29, 30syl2anc 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 ) ) )  ->  ( P  .\/  P )  =  P )
3214oveq2d 6107 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
3331, 32eqtr3d 2477 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  =  ( P  .\/  Q ) )
3433oveq1d 6106 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  ./\  W )  =  ( ( P  .\/  Q )  ./\  W )
)
3528, 34eqtrd 2475 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  (
( R  .\/  ( F `  R )
)  ./\  W )  =  ( ( P 
.\/  Q )  ./\  W ) )
36 cdlemef50.u . 2  |-  U  =  ( ( P  .\/  Q )  ./\  W )
3735, 36syl6eqr 2493 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  (
( R  .\/  ( F `  R )
)  ./\  W )  =  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   [_csb 3288   ifcif 3791   class class class wbr 4292    e. cmpt 4350   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   0.cp0 15207   Atomscatm 32908   HLchlt 32995   LHypclh 33628
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-lhyp 33632
This theorem is referenced by:  cdleme50trn123  34198
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