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Theorem dihglblem3N 34832
Description: Isomorphism H of a lattice glb. (Contributed by NM, 20-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 ) }
dihglblem.i  |-  J  =  ( ( DIsoB `  K
) `  W )
dihglblem.ih  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihglblem3N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  T ) )  = 
|^|_ x  e.  T  ( I `  x
) )
Distinct variable groups:    x, u, v,  ./\    x,  .<_    x, B, u    x, G    x, H    x, K    x, S, u, v    x, T    x, W, u, v    u,  .<_ , v   
v, B    u, G, v    u, H, v    u, K, v
Allowed substitution hints:    T( v, u)    I( x, v, u)    J( x, v, u)

Proof of Theorem dihglblem3N
StepHypRef Expression
1 simp1 1005 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dihglblem.t . . . . . 6  |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }
3 simp11l 1116 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  K  e.  HL )
4 hllat 32898 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 17 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  K  e.  Lat )
6 simp12l 1118 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  S  C_  B )
7 simp3 1007 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  v  e.  S )
86, 7sseldd 3465 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  v  e.  B )
9 simp11r 1117 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  W  e.  H )
10 dihglblem.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  K
)
11 dihglblem.h . . . . . . . . . . . . 13  |-  H  =  ( LHyp `  K
)
1210, 11lhpbase 33532 . . . . . . . . . . . 12  |-  ( W  e.  H  ->  W  e.  B )
139, 12syl 17 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  W  e.  B )
14 dihglblem.l . . . . . . . . . . . 12  |-  .<_  =  ( le `  K )
15 dihglblem.m . . . . . . . . . . . 12  |-  ./\  =  ( meet `  K )
1610, 14, 15latmle2 16322 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  v  e.  B  /\  W  e.  B )  ->  ( v  ./\  W
)  .<_  W )
175, 8, 13, 16syl3anc 1264 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  (
v  ./\  W )  .<_  W )
18173expia 1207 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B )  ->  (
v  e.  S  -> 
( v  ./\  W
)  .<_  W ) )
19 breq1 4426 . . . . . . . . . 10  |-  ( u  =  ( v  ./\  W )  ->  ( u  .<_  W  <->  ( v  ./\  W )  .<_  W )
)
2019biimprcd 228 . . . . . . . . 9  |-  ( ( v  ./\  W )  .<_  W  ->  ( u  =  ( v  ./\  W )  ->  u  .<_  W ) )
2118, 20syl6 34 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B )  ->  (
v  e.  S  -> 
( u  =  ( v  ./\  W )  ->  u  .<_  W )
) )
2221rexlimdv 2912 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B )  ->  ( E. v  e.  S  u  =  ( v  ./\  W )  ->  u  .<_  W ) )
2322ss2rabdv 3542 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }  C_  { u  e.  B  |  u  .<_  W } )
242, 23syl5eqss 3508 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  T  C_ 
{ u  e.  B  |  u  .<_  W }
)
25 dihglblem.i . . . . . . 7  |-  J  =  ( ( DIsoB `  K
) `  W )
2610, 14, 11, 25dibdmN 34694 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  J  =  {
u  e.  B  |  u  .<_  W } )
27263ad2ant1 1026 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  dom  J  =  { u  e.  B  |  u  .<_  W } )
2824, 27sseqtr4d 3501 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  T  C_ 
dom  J )
29 dihglblem.g . . . . . 6  |-  G  =  ( glb `  K
)
3010, 14, 15, 29, 11, 2dihglblem2aN 34830 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
31303adant3 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  T  =/=  (/) )
3229, 11, 25dibglbN 34703 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  C_  dom  J  /\  T  =/=  (/) ) )  ->  ( J `  ( G `  T ) )  = 
|^|_ x  e.  T  ( J `  x ) )
331, 28, 31, 32syl12anc 1262 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( J `  ( G `  T ) )  = 
|^|_ x  e.  T  ( J `  x ) )
3410, 14, 15, 29, 11, 2dihglblem2N 34831 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  B  /\  ( G `  S
)  .<_  W )  -> 
( G `  S
)  =  ( G `
 T ) )
35343adant2r 1259 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  =  ( G `  T ) )
3635fveq2d 5885 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( J `  ( G `  S ) )  =  ( J `  ( G `  T )
) )
37 simpl1 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
3824sselda 3464 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  x  e.  { u  e.  B  |  u  .<_  W }
)
39 breq1 4426 . . . . . . 7  |-  ( u  =  x  ->  (
u  .<_  W  <->  x  .<_  W ) )
4039elrab 3228 . . . . . 6  |-  ( x  e.  { u  e.  B  |  u  .<_  W }  <->  ( x  e.  B  /\  x  .<_  W ) )
4138, 40sylib 199 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  (
x  e.  B  /\  x  .<_  W ) )
42 dihglblem.ih . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
4310, 14, 11, 42, 25dihvalb 34774 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x  e.  B  /\  x  .<_  W ) )  ->  (
I `  x )  =  ( J `  x ) )
4437, 41, 43syl2anc 665 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  (
I `  x )  =  ( J `  x ) )
4544iineq2dv 4322 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  |^|_ x  e.  T  ( I `  x )  =  |^|_ x  e.  T  ( J `
 x ) )
4633, 36, 453eqtr4rd 2474 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  |^|_ x  e.  T  ( I `  x )  =  ( J `  ( G `
 S ) ) )
47 simp1l 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  K  e.  HL )
48 hlclat 32893 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
4947, 48syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  K  e.  CLat )
50 simp2l 1031 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  S  C_  B )
5110, 29clatglbcl 16359 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
5249, 50, 51syl2anc 665 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  e.  B )
53 simp3 1007 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  .<_  W )
5410, 14, 11, 42, 25dihvalb 34774 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( G `
 S )  e.  B  /\  ( G `
 S )  .<_  W ) )  -> 
( I `  ( G `  S )
)  =  ( J `
 ( G `  S ) ) )
551, 52, 53, 54syl12anc 1262 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( J `  ( G `  S )
) )
5635fveq2d 5885 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( I `  ( G `  T )
) )
5746, 55, 563eqtr2rd 2470 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  T ) )  = 
|^|_ x  e.  T  ( I `  x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   {crab 2775    C_ wss 3436   (/)c0 3761   |^|_ciin 4300   class class class wbr 4423   dom cdm 4853   ` cfv 5601  (class class class)co 6305   Basecbs 15120   lecple 15196   glbcglb 16187   meetcmee 16189   Latclat 16290   CLatccla 16352   HLchlt 32885   LHypclh 33518   DIsoBcdib 34675   DIsoHcdih 34765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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-mpt2 6310  df-map 7485  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-p1 16285  df-lat 16291  df-clat 16353  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886  df-lhyp 33522  df-laut 33523  df-ldil 33638  df-ltrn 33639  df-trl 33694  df-disoa 34566  df-dib 34676  df-dih 34766
This theorem is referenced by:  dihglblem3aN  34833
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