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Theorem cdleme50trn3 35642
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 999 . . . . 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 2467 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 cdlemef50.a . . . . . 6  |-  A  =  ( Atoms `  K )
7 cdlemef50.h . . . . . 6  |-  H  =  ( LHyp `  K
)
83, 4, 5, 6, 7lhpmat 35119 . . . . 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 34379 . . . . . . . . 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 35483 . . . . . . . 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 6310 . . . . . 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 1047 . . . . . . 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 34458 . . . . . . 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 2508 . . . . 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 6309 . . . 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 1000 . . . . 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 35119 . . . . 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 2518 . . 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 1049 . . . . . 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 34458 . . . . . 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 6310 . . . . 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 2510 . . . 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 6309 . . 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 2508 . 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 2526 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   [_csb 3440   ifcif 3944   class class class wbr 4452    |-> cmpt 4510   ` cfv 5593   iota_crio 6254  (class class class)co 6294   Basecbs 14502   lecple 14574   joincjn 15443   meetcmee 15444   0.cp0 15536   Atomscatm 34353   HLchlt 34440   LHypclh 35073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-poset 15445  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-covers 34356  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441  df-lhyp 35077
This theorem is referenced by:  cdleme50trn123  35643
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