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Theorem cdlemg2cex 30780
Description: Any translation is one of our  F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 30752? (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 30777 . 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 734 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  K  e.  HL )
7 simpllr 735 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  W  e.  H
)
8 simplrl 736 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  p  e.  A
)
9 simprl 732 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  -.  p  .<_  W )
10 simplrr 737 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  q  e.  A )
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  q  e.  A
)
11 simprr 733 . . . . . . 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 2283 . . . . . . . 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 30760 . . . . . . 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 1198 . . . . . 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 2294 . . . . 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 622 . . . 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 936 . . . 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 936 . . . 4  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  /\  F  =  G ) )
2623, 24, 253bitr4g 279 . . 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 2584 . 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 244 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   [_csb 3081   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  cdlemg2ce  30781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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