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Theorem cdlemg1fvawlemN 34536
Description: Lemma for ltrniotafvawN 34541. (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
cdlemg1fvawlemN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
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    R, 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)    R( f)    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 cdlemg1fvawlemN
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 ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme46fvaw 34464 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( G `  R )  e.  A  /\  -.  ( G `  R )  .<_  W ) )
12 cdlemg1.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemg1.f . . . . . . 7  |-  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13cdlemg1b2 34534 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  =  G )
1514adantr 465 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  F  =  G )
1615fveq1d 5796 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( F `  R )  =  ( G `  R ) )
1716eleq1d 2521 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( F `  R )  e.  A  <->  ( G `  R )  e.  A
) )
1816breq1d 4405 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( F `  R )  .<_  W  <->  ( G `  R )  .<_  W ) )
1918notbid 294 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( -.  ( F `  R ) 
.<_  W  <->  -.  ( G `  R )  .<_  W ) )
2017, 19anbi12d 710 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( (
( F `  R
)  e.  A  /\  -.  ( F `  R
)  .<_  W )  <->  ( ( G `  R )  e.  A  /\  -.  ( G `  R )  .<_  W ) ) )
2111, 20mpbird 232 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   [_csb 3390   ifcif 3894   class class class wbr 4395    |-> cmpt 4453   ` cfv 5521   iota_crio 6155  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-undef 6897  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by:  ltrniotafvawN  34541
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