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Theorem cdlemg2cN 34230
Description: Any translation belongs to the set of functions constructed for cdleme 34201. TODO: Fix comment. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
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 )
cdlemg2.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemg2.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemg2.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemg2.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
cdlemg2cN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  Q )  -> 
( F  e.  T  <->  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    P, s, t, x, y, z    Q, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z
Allowed substitution hints:    D( t)    T( x, y, z, t, s)    E( t, s)    F( x, y, z, t, s)    G( x, y, z, t, s)

Proof of Theorem cdlemg2cN
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, 4cdlemg1cN 34228 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  Q )  -> 
( F  e.  T  <->  F  =  ( iota_ f  e.  T  ( f `  P )  =  Q ) ) )
6 cdlemg2.b . . . . 5  |-  B  =  ( Base `  K
)
7 cdlemg2.j . . . . 5  |-  .\/  =  ( join `  K )
8 cdlemg2.m . . . . 5  |-  ./\  =  ( meet `  K )
9 cdlemg2.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
10 cdlemg2.d . . . . 5  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
11 cdlemg2.e . . . . 5  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
12 cdlemg2.g . . . . 5  |-  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 ) )
13 eqid 2441 . . . . 5  |-  ( iota_ f  e.  T  ( f `
 P )  =  Q )  =  (
iota_ f  e.  T  ( f `  P
)  =  Q )
146, 1, 7, 8, 2, 3, 9, 10, 11, 12, 4, 13cdlemg1b2 34212 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( iota_ f  e.  T  ( f `
 P )  =  Q )  =  G )
1514adantr 465 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  Q )  -> 
( iota_ f  e.  T  ( f `  P
)  =  Q )  =  G )
1615eqeq2d 2452 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  Q )  -> 
( F  =  (
iota_ f  e.  T  ( f `  P
)  =  Q )  <-> 
F  =  G ) )
175, 16bitrd 253 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `  P )  =  Q )  -> 
( F  e.  T  <->  F  =  G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   [_csb 3286   ifcif 3789   class class class wbr 4290    e. cmpt 4348   ` cfv 5416   iota_crio 6049  (class class class)co 6089   Basecbs 14172   lecple 14243   joincjn 15112   meetcmee 15113   Atomscatm 32905   HLchlt 32992   LHypclh 33625   LTrncltrn 33742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-riotaBAD 32601
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-undef 6790  df-map 7214  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-p1 15208  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993  df-llines 33139  df-lplanes 33140  df-lvols 33141  df-lines 33142  df-psubsp 33144  df-pmap 33145  df-padd 33437  df-lhyp 33629  df-laut 33630  df-ldil 33745  df-ltrn 33746  df-trl 33800
This theorem is referenced by:  cdlemg2dN  34231
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