Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg2cex Structured version   Unicode version

Theorem cdlemg2cex 35788
Description: Any translation is one of our  F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 35760? (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 35785 . 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 2467 . . . . . . . 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 35768 . . . . . . 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 1244 . . . . . 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 2481 . . . . 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 975 . . . 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 975 . . . 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 2984 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   [_csb 3440   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594   iota_crio 6255  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Atomscatm 34461   HLchlt 34548   LHypclh 35181   LTrncltrn 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-undef 7014  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356
This theorem is referenced by:  cdlemg2ce  35789
  Copyright terms: Public domain W3C validator