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Theorem cdlemg2cex 30849
Description: Any translation is one of our  F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 30821? (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 30846 . 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 2358 . . . . . . . 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 30829 . . . . . . 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 2369 . . . . 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 2660 . 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 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   LTrncltrn 30359
This theorem is referenced by:  cdlemg2ce  30850
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-map 6862  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  df-laut 30247  df-ldil 30362  df-ltrn 30363  df-trl 30417
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