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Theorem dihglblem3N 35263
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 988 . . . 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 1099 . . . . . . . . . . . 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 33331 . . . . . . . . . . . 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 1101 . . . . . . . . . . . 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 990 . . . . . . . . . . . 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 3464 . . . . . . . . . . 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 1100 . . . . . . . . . . . 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 33965 . . . . . . . . . . . 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 15365 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  v  e.  B  /\  W  e.  B )  ->  ( v  ./\  W
)  .<_  W )
175, 8, 13, 16syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  u  e.  B  /\  v  e.  S )  ->  (
v  ./\  W )  .<_  W )
18173expia 1190 . . . . . . . . 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 4402 . . . . . . . . . 10  |-  ( u  =  ( v  ./\  W )  ->  ( u  .<_  W  <->  ( v  ./\  W )  .<_  W )
)
2019biimprcd 225 . . . . . . . . 9  |-  ( ( v  ./\  W )  .<_  W  ->  ( u  =  ( v  ./\  W )  ->  u  .<_  W ) )
2118, 20syl6 33 . . . . . . . 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 2944 . . . . . . 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 3540 . . . . . 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 3507 . . . . 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 35125 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  J  =  {
u  e.  B  |  u  .<_  W } )
27263ad2ant1 1009 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  dom  J  =  { u  e.  B  |  u  .<_  W } )
2824, 27sseqtr4d 3500 . . . 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 35261 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) )  ->  T  =/=  (/) )
31303adant3 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  T  =/=  (/) )
3229, 11, 25dibglbN 35134 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  C_  dom  J  /\  T  =/=  (/) ) )  ->  ( J `  ( G `  T ) )  = 
|^|_ x  e.  T  ( J `  x ) )
331, 28, 31, 32syl12anc 1217 . . 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 35262 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  B  /\  ( G `  S
)  .<_  W )  -> 
( G `  S
)  =  ( G `
 T ) )
35343adant2r 1214 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  =  ( G `  T ) )
3635fveq2d 5802 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( J `  ( G `  S ) )  =  ( J `  ( G `  T )
) )
37 simpl1 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
3824sselda 3463 . . . . . 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 4402 . . . . . . 7  |-  ( u  =  x  ->  (
u  .<_  W  <->  x  .<_  W ) )
4039elrab 3222 . . . . . 6  |-  ( x  e.  { u  e.  B  |  u  .<_  W }  <->  ( x  e.  B  /\  x  .<_  W ) )
4138, 40sylib 196 . . . . 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 35205 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x  e.  B  /\  x  .<_  W ) )  ->  (
I `  x )  =  ( J `  x ) )
4437, 41, 43syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S )  .<_  W )  /\  x  e.  T )  ->  (
I `  x )  =  ( J `  x ) )
4544iineq2dv 4300 . . 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 2506 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  |^|_ x  e.  T  ( I `  x )  =  ( J `  ( G `
 S ) ) )
47 simp1l 1012 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  K  e.  HL )
48 hlclat 33326 . . . . 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 1014 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  S  C_  B )
5110, 29clatglbcl 15402 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
5249, 50, 51syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  e.  B )
53 simp3 990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  ( G `  S )  .<_  W )
5410, 14, 11, 42, 25dihvalb 35205 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( G `
 S )  e.  B  /\  ( G `
 S )  .<_  W ) )  -> 
( I `  ( G `  S )
)  =  ( J `
 ( G `  S ) ) )
551, 52, 53, 54syl12anc 1217 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( J `  ( G `  S )
) )
5635fveq2d 5802 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
.<_  W )  ->  (
I `  ( G `  S ) )  =  ( I `  ( G `  T )
) )
5746, 55, 563eqtr2rd 2502 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   {crab 2802    C_ wss 3435   (/)c0 3744   |^|_ciin 4279   class class class wbr 4399   dom cdm 4947   ` cfv 5525  (class class class)co 6199   Basecbs 14291   lecple 14363   glbcglb 15231   meetcmee 15233   Latclat 15333   CLatccla 15395   HLchlt 33318   LHypclh 33951   DIsoBcdib 35106   DIsoHcdih 35196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-map 7325  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126  df-disoa 34997  df-dib 35107  df-dih 35197
This theorem is referenced by:  dihglblem3aN  35264
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