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Theorem cdlemg2cex 31450
Description: Any translation is one of our  F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 31422? (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 31447 . 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 736 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  K  e.  HL )
7 simpllr 737 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  W  e.  H
)
8 simplrl 738 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  p  e.  A
)
9 simprl 734 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  -.  p  .<_  W )
10 simplrr 739 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  q  e.  A
)
11 simprr 735 . . . . . . 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 2438 . . . . . . . 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 31430 . . . . . . 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 1201 . . . . . 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 2449 . . . . 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 624 . . . 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 939 . . . 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 939 . . . 4  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  F  =  G ) )
2623, 24, 253bitr4g 281 . . 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 2748 . 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 246 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   [_csb 3253   ifcif 3741   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960
This theorem is referenced by:  cdlemg2ce  31451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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