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Theorem cdlemg1finvtrlemN 31069
Description: Lemma for ltrniotacnvN 31074. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
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
cdlemg1finvtrlemN  |-  ( ( ( 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 cdlemg1finvtrlemN
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 )
12 cdlemg1.f . . . 4  |-  F  =  ( iota_ f  e.  T
( f `  P
)  =  Q )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemg1b2 31065 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  =  G )
1413cnveqd 5015 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' F  =  `' G )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdleme51finvtrN 31052 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' G  e.  T )
1614, 15eqeltrd 2486 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' F  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   [_csb 3219   ifcif 3707   class class class wbr 4180    e. cmpt 4234   `'ccnv 4844   ` cfv 5421  (class class class)co 6048   iota_crio 6509   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Atomscatm 29758   HLchlt 29845   LHypclh 30478   LTrncltrn 30595
This theorem is referenced by:  ltrniotacnvN  31074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653
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