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Theorem cdleme50trn2 33534
Description: Part of proof that  F is a translation. Remove  S hypotheses no longer needed from cdleme50trn2a 33533. 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
cdleme50trn2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  ( ( 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 cdleme50trn2
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 simp11 1025 . . 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  .<_  ( P  .\/  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 1026 . . 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  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp13 1027 . . 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  .<_  ( P  .\/  Q ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp2l 1021 . . 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  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
5 cdlemef50.l . . . 4  |-  .<_  =  ( le `  K )
6 cdlemef50.j . . . 4  |-  .\/  =  ( join `  K )
7 cdlemef50.a . . . 4  |-  A  =  ( Atoms `  K )
8 cdlemef50.h . . . 4  |-  H  =  ( LHyp `  K
)
95, 6, 7, 8cdlemb2 33022 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. e  e.  A  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) )
101, 2, 3, 4, 9syl121anc 1233 . 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  .<_  ( P  .\/  Q ) )  ->  E. e  e.  A  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) )
11 simp1 995 . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
12 simp2l 1021 . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  ->  P  =/=  Q )
13 simp2r 1022 . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  -> 
( R  e.  A  /\  -.  R  .<_  W ) )
14 simp3rl 1068 . . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  -> 
e  e.  A )
15 simprrl 764 . . . . . . . . . 10  |-  ( ( R  .<_  ( P  .\/  Q )  /\  (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) ) )  ->  -.  e  .<_  W )
16153ad2ant3 1018 . . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  ->  -.  e  .<_  W )
1714, 16jca 530 . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  -> 
( e  e.  A  /\  -.  e  .<_  W ) )
18 simp3l 1023 . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  ->  R  .<_  ( P  .\/  Q ) )
19 simprrr 765 . . . . . . . . 9  |-  ( ( R  .<_  ( P  .\/  Q )  /\  (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) ) )  ->  -.  e  .<_  ( P 
.\/  Q ) )
20193ad2ant3 1018 . . . . . . . 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 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  ->  -.  e  .<_  ( P 
.\/  Q ) )
21 cdlemef50.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
22 cdlemef50.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
23 cdlemef50.u . . . . . . . . 9  |-  U  =  ( ( P  .\/  Q )  ./\  W )
24 cdlemef50.d . . . . . . . . 9  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
25 cdlemefs50.e . . . . . . . . 9  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
26 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 ) )
2721, 5, 6, 22, 7, 8, 23, 24, 25, 26cdleme50trn2a 33533 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( e  e.  A  /\  -.  e  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  e  .<_  ( P 
.\/  Q ) ) )  ->  ( ( R  .\/  ( F `  R ) )  ./\  W )  =  U )
2811, 12, 13, 17, 18, 20, 27syl132anc 1246 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( R 
.<_  ( P  .\/  Q
)  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) ) )  -> 
( ( R  .\/  ( F `  R ) )  ./\  W )  =  U )
29283exp 1194 . . . . . 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  .<_  ( P  .\/  Q )  /\  ( e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) ) ) )  ->  (
( R  .\/  ( F `  R )
)  ./\  W )  =  U ) ) )
3029exp4a 604 . . . . 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  .<_  ( P  .\/  Q
)  ->  ( (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  ( F `  R )
)  ./\  W )  =  U ) ) ) )
31303imp 1189 . . . 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  .<_  ( P  .\/  Q ) )  ->  ( (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  ( F `  R )
)  ./\  W )  =  U ) )
3231expd 434 . . 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  .<_  ( P  .\/  Q ) )  ->  ( e  e.  A  ->  ( ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) )  ->  ( ( R 
.\/  ( F `  R ) )  ./\  W )  =  U ) ) )
3332rexlimdv 2891 . 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  .<_  ( P  .\/  Q ) )  ->  ( E. e  e.  A  ( -.  e  .<_  W  /\  -.  e  .<_  ( P 
.\/  Q ) )  ->  ( ( R 
.\/  ( F `  R ) )  ./\  W )  =  U ) )
3410, 33mpd 15 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  .<_  ( P  .\/  Q ) )  ->  ( ( R  .\/  ( F `  R ) )  ./\  W )  =  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   A.wral 2751   E.wrex 2752   [_csb 3370   ifcif 3882   class class class wbr 4392    |-> cmpt 4450   ` cfv 5523   iota_crio 6193  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   Atomscatm 32245   HLchlt 32332   LHypclh 32965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-riotaBAD 31941
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-undef 6957  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479  df-lplanes 32480  df-lvols 32481  df-lines 32482  df-psubsp 32484  df-pmap 32485  df-padd 32777  df-lhyp 32969
This theorem is referenced by:  cdleme50trn12  33535
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