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Theorem cdleme50trn2 30809
Description: Part of proof that  F is a translation. Remove  S hypotheses no longer needed from cdleme50trn2a 30808. 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 985 . . 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 986 . . 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 987 . . 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 981 . . 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 30299 . . 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 1187 . 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 955 . . . . . . . 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 981 . . . . . . . 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 982 . . . . . . . 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 1028 . . . . . . . . 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 740 . . . . . . . . . 10  |-  ( ( R  .<_  ( P  .\/  Q )  /\  (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) ) )  ->  -.  e  .<_  W )
16153ad2ant3 978 . . . . . . . . 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 518 . . . . . . . 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 983 . . . . . . . 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 741 . . . . . . . . 9  |-  ( ( R  .<_  ( P  .\/  Q )  /\  (
e  e.  A  /\  ( -.  e  .<_  W  /\  -.  e  .<_  ( P  .\/  Q ) ) ) )  ->  -.  e  .<_  ( P 
.\/  Q ) )
20193ad2ant3 978 . . . . . . . 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 30808 . . . . . . . 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 1200 . . . . . . 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 1150 . . . . . 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 589 . . . . 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 1145 . . . 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 ) )
3231exp3a 425 . . 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 2742 . 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 14 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   [_csb 3157   ifcif 3641   class class class wbr 4104    e. cmpt 4158   ` cfv 5337  (class class class)co 5945   iota_crio 6384   Basecbs 13245   lecple 13312   joincjn 14177   meetcmee 14178   Atomscatm 29522   HLchlt 29609   LHypclh 30242
This theorem is referenced by:  cdleme50trn12  30810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-llines 29756  df-lplanes 29757  df-lvols 29758  df-lines 29759  df-psubsp 29761  df-pmap 29762  df-padd 30054  df-lhyp 30246
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