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Theorem dihglblem2aN 32105
Description: Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihglblem.b  |-  B  =  ( Base `  K
)
dihglblem.l  |-  .<_  =  ( le `  K )
dihglblem.m  |-  ./\  =  ( meet `  K )
dihglblem.g  |-  G  =  ( glb `  K
)
dihglblem.h  |-  H  =  ( LHyp `  K
)
dihglblem.t  |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }
Assertion
Ref Expression
dihglblem2aN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
Distinct variable groups:    v, u,  ./\    u, B    u, S, v   
u, W, v
Allowed substitution hints:    B( v)    T( v, u)    G( v, u)    H( v, u)    K( v, u)   
.<_ ( v, u)

Proof of Theorem dihglblem2aN
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihglblem.t . . 3  |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }
21a1i 10 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =  {
u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) } )
3 simprr 733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
4 n0 3477 . . . 4  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
53, 4sylib 188 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  E. z  z  e.  S )
6 hllat 30175 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
76ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  K  e.  Lat )
8 simplrl 736 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  S  C_  B )
9 simpr 447 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  z  e.  S )
108, 9sseldd 3194 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  z  e.  B )
11 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  W  e.  H )
12 dihglblem.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
13 dihglblem.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
1412, 13lhpbase 30809 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  B )
1511, 14syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  W  e.  B )
16 dihglblem.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
1712, 16latmcl 14173 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  W  e.  B )  ->  ( z  ./\  W
)  e.  B )
187, 10, 15, 17syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  e.  B )
19 eqidd 2297 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  =  ( z 
./\  W ) )
20 oveq1 5881 . . . . . . . . . 10  |-  ( v  =  z  ->  (
v  ./\  W )  =  ( z  ./\  W ) )
2120eqeq2d 2307 . . . . . . . . 9  |-  ( v  =  z  ->  (
( z  ./\  W
)  =  ( v 
./\  W )  <->  ( z  ./\  W )  =  ( z  ./\  W )
) )
2221rspcev 2897 . . . . . . . 8  |-  ( ( z  e.  S  /\  ( z  ./\  W
)  =  ( z 
./\  W ) )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
239, 19, 22syl2anc 642 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
24 ovex 5899 . . . . . . . 8  |-  ( z 
./\  W )  e. 
_V
25 eleq1 2356 . . . . . . . . 9  |-  ( w  =  ( z  ./\  W )  ->  ( w  e.  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  <->  ( z  ./\  W )  e.  {
u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) } ) )
26 eqeq1 2302 . . . . . . . . . . 11  |-  ( u  =  ( z  ./\  W )  ->  ( u  =  ( v  ./\  W )  <->  ( z  ./\  W )  =  ( v 
./\  W ) ) )
2726rexbidv 2577 . . . . . . . . . 10  |-  ( u  =  ( z  ./\  W )  ->  ( E. v  e.  S  u  =  ( v  ./\  W )  <->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) ) )
2827elrab 2936 . . . . . . . . 9  |-  ( ( z  ./\  W )  e.  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  <->  ( (
z  ./\  W )  e.  B  /\  E. v  e.  S  ( z  ./\  W )  =  ( v  ./\  W )
) )
2925, 28syl6bb 252 . . . . . . . 8  |-  ( w  =  ( z  ./\  W )  ->  ( w  e.  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  <->  ( (
z  ./\  W )  e.  B  /\  E. v  e.  S  ( z  ./\  W )  =  ( v  ./\  W )
) ) )
3024, 29spcev 2888 . . . . . . 7  |-  ( ( ( z  ./\  W
)  e.  B  /\  E. v  e.  S  ( z  ./\  W )  =  ( v  ./\  W ) )  ->  E. w  w  e.  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) } )
3118, 23, 30syl2anc 642 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  E. w  w  e. 
{ u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) } )
32 n0 3477 . . . . . 6  |-  ( { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  =/=  (/)  <->  E. w  w  e.  {
u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) } )
3331, 32sylibr 203 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  =/=  (/) )
3433ex 423 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  ( z  e.  S  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  =/=  (/) ) )
3534exlimdv 1626 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  ( E. z 
z  e.  S  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  =/=  (/) ) )
365, 35mpd 14 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W
) }  =/=  (/) )
372, 36eqnetrd 2477 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   glbcglb 14093   meetcmee 14095   Latclat 14167   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  dihglblem3N  32107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-lat 14168  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
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