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Theorem dihglblem2aN 32153
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 735 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
4 n0 3639 . . . 4  |-  ( S  =/=  (/)  <->  E. z  z  e.  S )
53, 4sylib 190 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  E. z  z  e.  S )
6 hllat 30223 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
76ad3antrrr 712 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  K  e.  Lat )
8 simplrl 738 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  S  C_  B )
9 simpr 449 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  z  e.  S )
108, 9sseldd 3351 . . . . . 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 30857 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
1413ad3antlr 713 . . . . . 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 14482 . . . . . 6  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  W  e.  B )  ->  ( z  ./\  W
)  e.  B )
177, 10, 14, 16syl3anc 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  e.  B )
18 eqidd 2439 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  =  ( z 
./\  W ) )
19 oveq1 6090 . . . . . . . 8  |-  ( v  =  z  ->  (
v  ./\  W )  =  ( z  ./\  W ) )
2019eqeq2d 2449 . . . . . . 7  |-  ( v  =  z  ->  (
( z  ./\  W
)  =  ( v 
./\  W )  <->  ( z  ./\  W )  =  ( z  ./\  W )
) )
2120rspcev 3054 . . . . . 6  |-  ( ( z  e.  S  /\  ( z  ./\  W
)  =  ( z 
./\  W ) )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
229, 18, 21syl2anc 644 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) )
23 ovex 6108 . . . . . 6  |-  ( z 
./\  W )  e. 
_V
24 eleq1 2498 . . . . . . 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 2444 . . . . . . . . 9  |-  ( u  =  ( z  ./\  W )  ->  ( u  =  ( v  ./\  W )  <->  ( z  ./\  W )  =  ( v 
./\  W ) ) )
2625rexbidv 2728 . . . . . . . 8  |-  ( u  =  ( z  ./\  W )  ->  ( E. v  e.  S  u  =  ( v  ./\  W )  <->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) ) )
2726elrab 3094 . . . . . . 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 254 . . . . . 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 3045 . . . . 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 644 . . . 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 3639 . . . 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 205 . . 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 1649 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W
) }  =/=  (/) )
342, 33eqnetrd 2621 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   glbcglb 14402   meetcmee 14404   Latclat 14476   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  dihglblem3N  32155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-lat 14477  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847
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