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Theorem cdleme50trn2 34553
Description: Part of proof that  F is a translation. Remove  S hypotheses no longer needed from cdleme50trn2a 34552. 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 1018 . . 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 1019 . . 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 1020 . . 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 1014 . . 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 34043 . . 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 1224 . 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 988 . . . . . . . 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 1014 . . . . . . . 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 1015 . . . . . . . 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 1061 . . . . . . . . 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 763 . . . . . . . . . 10  |-  ( ( R  .<_  ( P  .\/  Q )  /\  (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) ) )  ->  -.  e  .<_  W )
16153ad2ant3 1011 . . . . . . . . 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 532 . . . . . . . 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 1016 . . . . . . . 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 764 . . . . . . . . 9  |-  ( ( R  .<_  ( P  .\/  Q )  /\  (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) ) )  ->  -.  e  .<_  ( P 
.\/  Q ) )
20193ad2ant3 1011 . . . . . . . 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 34552 . . . . . . . 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 1237 . . . . . . 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 1187 . . . . . 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 606 . . . . 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 1182 . . . 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 436 . . 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 2946 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   [_csb 3398   ifcif 3902   class class class wbr 4403    |-> cmpt 4461   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   meetcmee 15237   Atomscatm 33266   HLchlt 33353   LHypclh 33986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-undef 6905  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501  df-lvols 33502  df-lines 33503  df-psubsp 33505  df-pmap 33506  df-padd 33798  df-lhyp 33990
This theorem is referenced by:  cdleme50trn12  34554
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