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Theorem dihglblem2aN 34938
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 11 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =  {
u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) } )
3 simprr 756 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
4 n0 3646 . . . 4  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
53, 4sylib 196 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  E. z  z  e.  S )
6 hllat 33008 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
76ad3antrrr 729 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  K  e.  Lat )
8 simplrl 759 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  S  C_  B )
9 simpr 461 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  z  e.  S )
108, 9sseldd 3357 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  z  e.  B )
11 dihglblem.b . . . . . . . 8  |-  B  =  ( Base `  K
)
12 dihglblem.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
1311, 12lhpbase 33642 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
1413ad3antlr 730 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  W  e.  B )
15 dihglblem.m . . . . . . 7  |-  ./\  =  ( meet `  K )
1611, 15latmcl 15222 . . . . . 6  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  W  e.  B )  ->  ( z  ./\  W
)  e.  B )
177, 10, 14, 16syl3anc 1218 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  e.  B )
18 eqidd 2444 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  =  ( z 
./\  W ) )
19 oveq1 6098 . . . . . . . 8  |-  ( v  =  z  ->  (
v  ./\  W )  =  ( z  ./\  W ) )
2019eqeq2d 2454 . . . . . . 7  |-  ( v  =  z  ->  (
( z  ./\  W
)  =  ( v 
./\  W )  <->  ( z  ./\  W )  =  ( z  ./\  W )
) )
2120rspcev 3073 . . . . . 6  |-  ( ( z  e.  S  /\  ( z  ./\  W
)  =  ( z 
./\  W ) )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
229, 18, 21syl2anc 661 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
23 ovex 6116 . . . . . 6  |-  ( z 
./\  W )  e. 
_V
24 eleq1 2503 . . . . . . 7  |-  ( 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 ) } ) )
25 eqeq1 2449 . . . . . . . . 9  |-  ( u  =  ( z  ./\  W )  ->  ( u  =  ( v  ./\  W )  <->  ( z  ./\  W )  =  ( v 
./\  W ) ) )
2625rexbidv 2736 . . . . . . . 8  |-  ( u  =  ( z  ./\  W )  ->  ( E. v  e.  S  u  =  ( v  ./\  W )  <->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) ) )
2726elrab 3117 . . . . . . 7  |-  ( ( 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 )
) )
2824, 27syl6bb 261 . . . . . 6  |-  ( 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 )
) ) )
2923, 28spcev 3064 . . . . 5  |-  ( ( ( 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 ) } )
3017, 22, 29syl2anc 661 . . . 4  |-  ( ( ( ( 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 ) } )
31 n0 3646 . . . 4  |-  ( { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  =/=  (/)  <->  E. w  w  e.  {
u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) } )
3230, 31sylibr 212 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  =/=  (/) )
335, 32exlimddv 1692 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W
) }  =/=  (/) )
342, 33eqnetrd 2626 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2606   E.wrex 2716   {crab 2719    C_ wss 3328   (/)c0 3637   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   glbcglb 15113   meetcmee 15115   Latclat 15215   HLchlt 32995   LHypclh 33628
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-lat 15216  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-lhyp 33632
This theorem is referenced by:  dihglblem3N  34940
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