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Theorem cdlemg1a 36439
Description: Shorter expression for  G. TODO: fix comment. TODO: shorten using cdleme 36429 or vice-versa? Also, if not shortened with cdleme 36429, then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-Apr-2013.)
Hypotheses
Ref Expression
cdlemg1.b  |-  B  =  ( Base `  K
)
cdlemg1.l  |-  .<_  =  ( le `  K )
cdlemg1.j  |-  .\/  =  ( join `  K )
cdlemg1.m  |-  ./\  =  ( meet `  K )
cdlemg1.a  |-  A  =  ( Atoms `  K )
cdlemg1.h  |-  H  =  ( LHyp `  K
)
cdlemg1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemg1.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemg1.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemg1.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 ) )
cdlemg1.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg1a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  G  =  ( iota_ f  e.  T  ( f `  P
)  =  Q ) )
Distinct variable groups:    t, s, x, y, z, A, f    B, f, s, t, x, y, z    D, f, s, x, y, z   
f, E, x, y, z    H, s, t, x, y, z    .\/ , f,
s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , f,
s, t, x, y, z    P, s, t, x, y, z    Q, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    A, f   
f, H    f, K    .<_ , f    P, f    Q, f    T, f    f, W    f, G
Allowed substitution hints:    D( t)    T( x, y, z, t, s)    U( f)    E( t, s)    G( x, y, z, t, s)

Proof of Theorem cdlemg1a
StepHypRef Expression
1 cdlemg1.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemg1.l . . . 4  |-  .<_  =  ( le `  K )
3 cdlemg1.j . . . 4  |-  .\/  =  ( join `  K )
4 cdlemg1.m . . . 4  |-  ./\  =  ( meet `  K )
5 cdlemg1.a . . . 4  |-  A  =  ( Atoms `  K )
6 cdlemg1.h . . . 4  |-  H  =  ( LHyp `  K
)
7 cdlemg1.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdlemg1.d . . . 4  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdlemg1.e . . . 4  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
10 cdlemg1.g . . . 4  |-  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 ) )
11 cdlemg1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdleme50ltrn 36426 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  G  e.  T )
13 simpll1 1035 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  ( K  e.  HL  /\  W  e.  H ) )
14 simplr 755 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  f  e.  T
)
1512ad2antrr 725 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  G  e.  T
)
16 simpll2 1036 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
17 simpr 461 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  ( f `  P )  =  Q )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme17d 36367 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( G `  P )  =  Q )
1918ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  ( G `  P )  =  Q )
2017, 19eqtr4d 2501 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  ( f `  P )  =  ( G `  P ) )
212, 5, 6, 11cdlemd 36075 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( f `  P )  =  ( G `  P ) )  ->  f  =  G )
2213, 14, 15, 16, 20, 21syl311anc 1242 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  /\  ( f `  P
)  =  Q )  ->  f  =  G )
2322ex 434 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  ->  ( ( f `  P )  =  Q  ->  f  =  G ) )
2418adantr 465 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  ->  ( G `  P
)  =  Q )
25 fveq1 5871 . . . . . 6  |-  ( f  =  G  ->  (
f `  P )  =  ( G `  P ) )
2625eqeq1d 2459 . . . . 5  |-  ( f  =  G  ->  (
( f `  P
)  =  Q  <->  ( G `  P )  =  Q ) )
2724, 26syl5ibrcom 222 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  ->  ( f  =  G  ->  ( f `  P )  =  Q ) )
2823, 27impbid 191 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  f  e.  T )  ->  ( ( f `  P )  =  Q  <-> 
f  =  G ) )
2912, 28riota5 6283 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( iota_ f  e.  T  ( f `
 P )  =  Q )  =  G )
3029eqcomd 2465 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  G  =  ( iota_ f  e.  T  ( f `  P
)  =  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   [_csb 3430   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   meetcmee 15701   Atomscatm 35131   HLchlt 35218   LHypclh 35851   LTrncltrn 35968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027
This theorem is referenced by:  cdlemg1b2  36440
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