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Theorem cdlemg2cex 34240
Description: Any translation is one of our  F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 34212? (Contributed by NM, 22-Apr-2013.)
Hypotheses
Ref Expression
cdlemg2.b  |-  B  =  ( Base `  K
)
cdlemg2.l  |-  .<_  =  ( le `  K )
cdlemg2.j  |-  .\/  =  ( join `  K )
cdlemg2.m  |-  ./\  =  ( meet `  K )
cdlemg2.a  |-  A  =  ( Atoms `  K )
cdlemg2.h  |-  H  =  ( LHyp `  K
)
cdlemg2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg2ex.u  |-  U  =  ( ( p  .\/  q )  ./\  W
)
cdlemg2ex.d  |-  D  =  ( ( t  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )
cdlemg2ex.e  |-  E  =  ( ( p  .\/  q )  ./\  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) ) )
cdlemg2ex.g  |-  G  =  ( 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
cdlemg2cex  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( F  e.  T  <->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )
) )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    .\/ , s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    q, p, A    F, p, q    H, p, q    K, p, q    .<_ , p, q    T, p, q    W, p, q, s, t, x, y, z
Allowed substitution hints:    B( q, p)    D( t, q, p)    T( x, y, z, t, s)    U( q, p)    E( t,
s, q, p)    F( x, y, z, t, s)    G( x, y, z, t, s, q, p)    .\/ ( q, p)   
./\ ( q, p)

Proof of Theorem cdlemg2cex
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cdlemg2.l . . 3  |-  .<_  =  ( le `  K )
2 cdlemg2.a . . 3  |-  A  =  ( Atoms `  K )
3 cdlemg2.h . . 3  |-  H  =  ( LHyp `  K
)
4 cdlemg2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
51, 2, 3, 4cdlemg1cex 34237 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( F  e.  T  <->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  ( iota_ f  e.  T  ( f `
 p )  =  q ) ) ) )
6 simplll 757 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  K  e.  HL )
7 simpllr 758 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  W  e.  H
)
8 simplrl 759 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  p  e.  A
)
9 simprl 755 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  -.  p  .<_  W )
10 simplrr 760 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  q  e.  A
)
11 simprr 756 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  -.  q  .<_  W )
12 cdlemg2.b . . . . . . . 8  |-  B  =  ( Base `  K
)
13 cdlemg2.j . . . . . . . 8  |-  .\/  =  ( join `  K )
14 cdlemg2.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
15 cdlemg2ex.u . . . . . . . 8  |-  U  =  ( ( p  .\/  q )  ./\  W
)
16 cdlemg2ex.d . . . . . . . 8  |-  D  =  ( ( t  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )
17 cdlemg2ex.e . . . . . . . 8  |-  E  =  ( ( p  .\/  q )  ./\  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) ) )
18 cdlemg2ex.g . . . . . . . 8  |-  G  =  ( 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 ) )
19 eqid 2443 . . . . . . . 8  |-  ( iota_ f  e.  T  ( f `
 p )  =  q )  =  (
iota_ f  e.  T  ( f `  p
)  =  q )
2012, 1, 13, 14, 2, 3, 15, 16, 17, 18, 4, 19cdlemg1b2 34220 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p  .<_  W )  /\  (
q  e.  A  /\  -.  q  .<_  W ) )  ->  ( iota_ f  e.  T  ( f `
 p )  =  q )  =  G )
216, 7, 8, 9, 10, 11, 20syl222anc 1234 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  ( iota_ f  e.  T  ( f `  p )  =  q )  =  G )
2221eqeq2d 2454 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  ( F  =  ( iota_ f  e.  T  ( f `  p
)  =  q )  <-> 
F  =  G ) )
2322pm5.32da 641 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  F  =  (
iota_ f  e.  T  ( f `  p
)  =  q ) )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  F  =  G ) ) )
24 df-3an 967 . . . 4  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  ( iota_ f  e.  T  ( f `
 p )  =  q ) )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  F  =  (
iota_ f  e.  T  ( f `  p
)  =  q ) ) )
25 df-3an 967 . . . 4  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  F  =  G ) )
2623, 24, 253bitr4g 288 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  ( iota_ f  e.  T  ( f `  p
)  =  q ) )  <->  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) ) )
27262rexbidva 2761 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  ( iota_ f  e.  T  ( f `  p
)  =  q ) )  <->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) ) )
285, 27bitrd 253 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( F  e.  T  <->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   [_csb 3293   ifcif 3796   class class class wbr 4297    e. cmpt 4355   ` cfv 5423   iota_crio 6056  (class class class)co 6096   Basecbs 14179   lecple 14250   joincjn 15119   meetcmee 15120   Atomscatm 32913   HLchlt 33000   LHypclh 33633   LTrncltrn 33750
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-riotaBAD 32609
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-undef 6797  df-map 7221  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-lines 33150  df-psubsp 33152  df-pmap 33153  df-padd 33445  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808
This theorem is referenced by:  cdlemg2ce  34241
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