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Theorem dihglblem3N 34571
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 32637 . . . . . . . . . . . 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 3471 . . . . . . . . . . 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 33271 . . . . . . . . . . . 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 16274 . . . . . . . . . . 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 4429 . . . . . . . . . 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 2922 . . . . . . 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 3548 . . . . . 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 3514 . . . . 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 34433 . . . . . 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 3507 . . . 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 34569 . . . . 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 34442 . . . 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 34570 . . . . 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 3470 . . . . . 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 4429 . . . . . . 7  |-  ( u  =  x  ->  (
u  .<_  W  <->  x  .<_  W ) )
4039elrab 3235 . . . . . 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 34513 . . . . 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 4325 . . 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 2481 . 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 32632 . . . . 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 16311 . . . 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 34513 . . 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 2477 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 1870    =/= wne 2625   E.wrex 2783   {crab 2786    C_ wss 3442   (/)c0 3767   |^|_ciin 4303   class class class wbr 4426   dom cdm 4854   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   glbcglb 16139   meetcmee 16141   Latclat 16242   CLatccla 16304   HLchlt 32624   LHypclh 33257   DIsoBcdib 34414   DIsoHcdih 34504
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-iin 4305  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-mpt2 6310  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32450  df-ol 32452  df-oml 32453  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-lhyp 33261  df-laut 33262  df-ldil 33377  df-ltrn 33378  df-trl 33433  df-disoa 34305  df-dib 34415  df-dih 34505
This theorem is referenced by:  dihglblem3aN  34572
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