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Theorem cdlemg1ltrnlem 35771
Description: Lemma for ltrniotacl 35776. (Contributed by NM, 18-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 )
cdlemg1.f  |-  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q )
Assertion
Ref Expression
cdlemg1ltrnlem  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
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)    F( x, y, z, t, f, s)    G( x, y, z, t, s)

Proof of Theorem cdlemg1ltrnlem
StepHypRef Expression
1 cdlemg1.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemg1.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemg1.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemg1.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemg1.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemg1.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemg1.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdlemg1.d . . 3  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdlemg1.e . . 3  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
10 cdlemg1.g . . 3  |-  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 . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
12 cdlemg1.f . . 3  |-  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemg1b2 35768 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  =  G )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdleme50ltrn 35754 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  G  e.  T )
1513, 14eqeltrd 2555 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   [_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:  ltrniotacl  35776
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