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Theorem dihglblem3N 31461
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 957 . . . 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 1068 . . . . . . . . . . . 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 29529 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . . . . . . . . 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 1070 . . . . . . . . . . . 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 959 . . . . . . . . . . . 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 3285 . . . . . . . . . . 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 1069 . . . . . . . . . . . 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 30163 . . . . . . . . . . . 12  |-  ( W  e.  H  ->  W  e.  B )
139, 12syl 16 . . . . . . . . . . 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 14426 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  v  e.  B  /\  W  e.  B )  ->  ( v  ./\  W
)  .<_  W )
175, 8, 13, 16syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  (
v  ./\  W )  .<_  W )
18173expia 1155 . . . . . . . . 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 4149 . . . . . . . . . 10  |-  ( u  =  ( v  ./\  W )  ->  ( u  .<_  W  <->  ( v  ./\  W )  .<_  W )
)
2019biimprcd 217 . . . . . . . . 9  |-  ( ( v  ./\  W )  .<_  W  ->  ( u  =  ( v  ./\  W )  ->  u  .<_  W ) )
2118, 20syl6 31 . . . . . . . 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 2765 . . . . . . 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 3360 . . . . . 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 3328 . . . . 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 31323 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  J  =  {
u  e.  B  |  u  .<_  W } )
27263ad2ant1 978 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  dom  J  =  { u  e.  B  |  u  .<_  W } )
2824, 27sseqtr4d 3321 . . . 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 31459 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
31303adant3 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  T  =/=  (/) )
3229, 11, 25dibglbN 31332 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  C_  dom  J  /\  T  =/=  (/) ) )  ->  ( J `  ( G `  T ) )  = 
|^|_ x  e.  T  ( J `  x ) )
331, 28, 31, 32syl12anc 1182 . . 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 31460 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  B  /\  ( G `  S
)  .<_  W )  -> 
( G `  S
)  =  ( G `
 T ) )
35343adant2r 1179 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  =  ( G `  T ) )
3635fveq2d 5665 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( J `  ( G `  S ) )  =  ( J `  ( G `  T )
) )
37 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
3824sselda 3284 . . . . . 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 4149 . . . . . . 7  |-  ( u  =  x  ->  (
u  .<_  W  <->  x  .<_  W ) )
4039elrab 3028 . . . . . 6  |-  ( x  e.  { u  e.  B  |  u  .<_  W }  <->  ( x  e.  B  /\  x  .<_  W ) )
4138, 40sylib 189 . . . . 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 31403 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x  e.  B  /\  x  .<_  W ) )  ->  (
I `  x )  =  ( J `  x ) )
4437, 41, 43syl2anc 643 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  (
I `  x )  =  ( J `  x ) )
4544iineq2dv 4050 . . 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 2423 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  |^|_ x  e.  T  ( I `  x )  =  ( J `  ( G `
 S ) ) )
47 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  K  e.  HL )
48 hlclat 29524 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
4947, 48syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  K  e.  CLat )
50 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  S  C_  B )
5110, 29clatglbcl 14461 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
5249, 50, 51syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  e.  B )
53 simp3 959 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  .<_  W )
5410, 14, 11, 42, 25dihvalb 31403 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( G `
 S )  e.  B  /\  ( G `
 S )  .<_  W ) )  -> 
( I `  ( G `  S )
)  =  ( J `
 ( G `  S ) ) )
551, 52, 53, 54syl12anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( J `  ( G `  S )
) )
5635fveq2d 5665 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( I `  ( G `  T )
) )
5746, 55, 563eqtr2rd 2419 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   {crab 2646    C_ wss 3256   (/)c0 3564   |^|_ciin 4029   class class class wbr 4146   dom cdm 4811   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   glbcglb 14320   meetcmee 14322   Latclat 14394   CLatccla 14456   HLchlt 29516   LHypclh 30149   DIsoBcdib 31304   DIsoHcdih 31394
This theorem is referenced by:  dihglblem3aN  31462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324  df-disoa 31195  df-dib 31305  df-dih 31395
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