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Theorem dihglblem2aN 36307
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 3794 . . . 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 34377 . . . . . . 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 3505 . . . . . 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 35011 . . . . . . 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 15542 . . . . . 6  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  W  e.  B )  ->  ( z  ./\  W
)  e.  B )
177, 10, 14, 16syl3anc 1228 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  e.  B )
18 eqidd 2468 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  /\  z  e.  S )  ->  ( z  ./\  W
)  =  ( z 
./\  W ) )
19 oveq1 6292 . . . . . . . 8  |-  ( v  =  z  ->  (
v  ./\  W )  =  ( z  ./\  W ) )
2019eqeq2d 2481 . . . . . . 7  |-  ( v  =  z  ->  (
( z  ./\  W
)  =  ( v 
./\  W )  <->  ( z  ./\  W )  =  ( z  ./\  W )
) )
2120rspcev 3214 . . . . . 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 6310 . . . . . 6  |-  ( z 
./\  W )  e. 
_V
24 eleq1 2539 . . . . . . 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 2471 . . . . . . . . 9  |-  ( u  =  ( z  ./\  W )  ->  ( u  =  ( v  ./\  W )  <->  ( z  ./\  W )  =  ( v 
./\  W ) ) )
2625rexbidv 2973 . . . . . . . 8  |-  ( u  =  ( z  ./\  W )  ->  ( E. v  e.  S  u  =  ( v  ./\  W )  <->  E. v  e.  S  ( z  ./\  W
)  =  ( v 
./\  W ) ) )
2726elrab 3261 . . . . . . 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 3205 . . . . 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 3794 . . . 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 1702 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W
) }  =/=  (/) )
342, 33eqnetrd 2760 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818    C_ wss 3476   (/)c0 3785   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   glbcglb 15433   meetcmee 15435   Latclat 15535   HLchlt 34364   LHypclh 34997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-lat 15536  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-lhyp 35001
This theorem is referenced by:  dihglblem3N  36309
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