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Theorem dihglblem3aN 34296
Description: Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-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
dihglblem3aN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  = 
|^|_ 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 dihglblem3aN
StepHypRef Expression
1 dihglblem.b . . . . 5  |-  B  =  ( Base `  K
)
2 dihglblem.l . . . . 5  |-  .<_  =  ( le `  K )
3 dihglblem.m . . . . 5  |-  ./\  =  ( meet `  K )
4 dihglblem.g . . . . 5  |-  G  =  ( glb `  K
)
5 dihglblem.h . . . . 5  |-  H  =  ( LHyp `  K
)
6 dihglblem.t . . . . 5  |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\  W ) }
71, 2, 3, 4, 5, 6dihglblem2N 34294 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  B  /\  ( G `  S
)  .<_  W )  -> 
( G `  S
)  =  ( G `
 T ) )
873adant2r 1225 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  =  ( G `  T ) )
98fveq2d 5852 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( I `  ( G `  T )
) )
10 dihglblem.i . . 3  |-  J  =  ( ( DIsoB `  K
) `  W )
11 dihglblem.ih . . 3  |-  I  =  ( ( DIsoH `  K
) `  W )
121, 2, 3, 4, 5, 6, 10, 11dihglblem3N 34295 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  T ) )  = 
|^|_ x  e.  T  ( I `  x
) )
139, 12eqtrd 2443 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  = 
|^|_ x  e.  T  ( I `  x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   {crab 2757    C_ wss 3413   (/)c0 3737   |^|_ciin 4271   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   glbcglb 15894   meetcmee 15896   HLchlt 32348   LHypclh 32981   DIsoBcdib 34138   DIsoHcdih 34228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-disoa 34029  df-dib 34139  df-dih 34229
This theorem is referenced by: (None)
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