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Theorem dihglblem2aN 34570
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 764 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
4 n0 3777 . . . 4  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
53, 4sylib 199 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  E. z  z  e.  S )
6 hllat 32638 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
76ad3antrrr 734 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  K  e.  Lat )
8 simplrl 768 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  S  C_  B )
9 simpr 462 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  z  e.  S )
108, 9sseldd 3471 . . . . . 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 33272 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
1413ad3antlr 735 . . . . . 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 16249 . . . . . 6  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  W  e.  B )  ->  ( z  ./\  W
)  e.  B )
177, 10, 14, 16syl3anc 1264 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  e.  B )
18 eqidd 2430 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  =  ( z 
./\  W ) )
19 oveq1 6312 . . . . . . . 8  |-  ( v  =  z  ->  (
v  ./\  W )  =  ( z  ./\  W ) )
2019eqeq2d 2443 . . . . . . 7  |-  ( v  =  z  ->  (
( z  ./\  W
)  =  ( v 
./\  W )  <->  ( z  ./\  W )  =  ( z  ./\  W )
) )
2120rspcev 3188 . . . . . 6  |-  ( ( z  e.  S  /\  ( z  ./\  W
)  =  ( z 
./\  W ) )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
229, 18, 21syl2anc 665 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
23 ovex 6333 . . . . . 6  |-  ( z 
./\  W )  e. 
_V
24 eleq1 2501 . . . . . . 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 2433 . . . . . . . . 9  |-  ( u  =  ( z  ./\  W )  ->  ( u  =  ( v  ./\  W )  <->  ( z  ./\  W )  =  ( v 
./\  W ) ) )
2625rexbidv 2946 . . . . . . . 8  |-  ( u  =  ( z  ./\  W )  ->  ( E. v  e.  S  u  =  ( v  ./\  W )  <->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) ) )
2726elrab 3235 . . . . . . 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 264 . . . . . 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 3179 . . . . 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 665 . . . 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 3777 . . . 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 215 . . 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 1773 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W
) }  =/=  (/) )
342, 33eqnetrd 2724 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870    =/= wne 2625   E.wrex 2783   {crab 2786    C_ wss 3442   (/)c0 3767   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   glbcglb 16139   meetcmee 16141   Latclat 16242   HLchlt 32625   LHypclh 33258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-lat 16243  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-lhyp 33262
This theorem is referenced by:  dihglblem3N  34572
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