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Theorem cdlemg1finvtrlemN 31446
Description: Lemma for ltrniotacnvN 31451. (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 31442 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  =  G )
1413cnveqd 5051 . 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 31429 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' G  e.  T )
1614, 15eqeltrd 2512 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   [_csb 3253   ifcif 3741   class class class wbr 4215    e. cmpt 4269   `'ccnv 4880   ` cfv 5457  (class class class)co 6084   iota_crio 6545   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LHypclh 30855   LTrncltrn 30972
This theorem is referenced by:  ltrniotacnvN  31451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030
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