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Theorem dihglblem3N 31485
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 955 . . . 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 1066 . . . . . . . . . . . 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 29553 . . . . . . . . . . . 12  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 15 . . . . . . . . . . 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 1068 . . . . . . . . . . . 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 957 . . . . . . . . . . . 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 3181 . . . . . . . . . . 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 1067 . . . . . . . . . . . 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 30187 . . . . . . . . . . . 12  |-  ( W  e.  H  ->  W  e.  B )
139, 12syl 15 . . . . . . . . . . 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 14183 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  v  e.  B  /\  W  e.  B )  ->  ( v  ./\  W
)  .<_  W )
175, 8, 13, 16syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  (
v  ./\  W )  .<_  W )
18173expia 1153 . . . . . . . . 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 4026 . . . . . . . . . 10  |-  ( u  =  ( v  ./\  W )  ->  ( u  .<_  W  <->  ( v  ./\  W )  .<_  W )
)
2019biimprcd 216 . . . . . . . . 9  |-  ( ( v  ./\  W )  .<_  W  ->  ( u  =  ( v  ./\  W )  ->  u  .<_  W ) )
2118, 20syl6 29 . . . . . . . 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 2666 . . . . . . 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 3254 . . . . . 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 3222 . . . . 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 31347 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  J  =  {
u  e.  B  |  u  .<_  W } )
27263ad2ant1 976 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  dom  J  =  { u  e.  B  |  u  .<_  W } )
2824, 27sseqtr4d 3215 . . . 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 31483 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
31303adant3 975 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  T  =/=  (/) )
3229, 11, 25dibglbN 31356 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  C_  dom  J  /\  T  =/=  (/) ) )  ->  ( J `  ( G `  T ) )  = 
|^|_ x  e.  T  ( J `  x ) )
331, 28, 31, 32syl12anc 1180 . . 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 31484 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  B  /\  ( G `  S
)  .<_  W )  -> 
( G `  S
)  =  ( G `
 T ) )
35343adant2r 1177 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  =  ( G `  T ) )
3635fveq2d 5529 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( J `  ( G `  S ) )  =  ( J `  ( G `  T )
) )
37 simpl1 958 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
3824sselda 3180 . . . . . 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 4026 . . . . . . 7  |-  ( u  =  x  ->  (
u  .<_  W  <->  x  .<_  W ) )
4039elrab 2923 . . . . . 6  |-  ( x  e.  { u  e.  B  |  u  .<_  W }  <->  ( x  e.  B  /\  x  .<_  W ) )
4138, 40sylib 188 . . . . 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 31427 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x  e.  B  /\  x  .<_  W ) )  ->  (
I `  x )  =  ( J `  x ) )
4437, 41, 43syl2anc 642 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  (
I `  x )  =  ( J `  x ) )
4544iineq2dv 3927 . . 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 2326 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  |^|_ x  e.  T  ( I `  x )  =  ( J `  ( G `
 S ) ) )
47 simp1l 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  K  e.  HL )
48 hlclat 29548 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
4947, 48syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  K  e.  CLat )
50 simp2l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  S  C_  B )
5110, 29clatglbcl 14218 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
5249, 50, 51syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  e.  B )
53 simp3 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  .<_  W )
5410, 14, 11, 42, 25dihvalb 31427 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( G `
 S )  e.  B  /\  ( G `
 S )  .<_  W ) )  -> 
( I `  ( G `  S )
)  =  ( J `
 ( G `  S ) ) )
551, 52, 53, 54syl12anc 1180 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( J `  ( G `  S )
) )
5635fveq2d 5529 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( I `  ( G `  T )
) )
5746, 55, 563eqtr2rd 2322 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   |^|_ciin 3906   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   glbcglb 14077   meetcmee 14079   Latclat 14151   CLatccla 14213   HLchlt 29540   LHypclh 30173   DIsoBcdib 31328   DIsoHcdih 31418
This theorem is referenced by:  dihglblem3aN  31486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-disoa 31219  df-dib 31329  df-dih 31419
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