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Theorem dia0 34793
Description: The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
dia0.b  |-  B  =  ( Base `  K
)
dia0.z  |-  .0.  =  ( 0. `  K )
dia0.h  |-  H  =  ( LHyp `  K
)
dia0.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
dia0  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { (  _I  |`  B ) } )

Proof of Theorem dia0
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 hlatl 33101 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
3 dia0.b . . . . . 6  |-  B  =  ( Base `  K
)
4 dia0.z . . . . . 6  |-  .0.  =  ( 0. `  K )
53, 4atl0cl 33044 . . . . 5  |-  ( K  e.  AtLat  ->  .0.  e.  B )
62, 5syl 16 . . . 4  |-  ( K  e.  HL  ->  .0.  e.  B )
76adantr 465 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  B )
8 dia0.h . . . . 5  |-  H  =  ( LHyp `  K
)
93, 8lhpbase 33738 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
10 eqid 2443 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
113, 10, 4atl0le 33045 . . . 4  |-  ( ( K  e.  AtLat  /\  W  e.  B )  ->  .0.  ( le `  K ) W )
122, 9, 11syl2an 477 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  ( le `  K ) W )
13 eqid 2443 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
14 eqid 2443 . . . 4  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
15 dia0.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
163, 10, 8, 13, 14, 15diaval 34773 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  .0.  e.  B  /\  .0.  ( le
`  K ) W ) )  ->  (
I `  .0.  )  =  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f ) ( le
`  K )  .0. 
} )
171, 7, 12, 16syl12anc 1216 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f ) ( le
`  K )  .0. 
} )
182ad2antrr 725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( ( LTrn `  K
) `  W )
)  ->  K  e.  AtLat
)
193, 8, 13, 14trlcl 33904 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
( trL `  K
) `  W ) `  f )  e.  B
)
203, 10, 4atlle0 33046 . . . . 5  |-  ( ( K  e.  AtLat  /\  (
( ( trL `  K
) `  W ) `  f )  e.  B
)  ->  ( (
( ( trL `  K
) `  W ) `  f ) ( le
`  K )  .0.  <->  ( ( ( trL `  K
) `  W ) `  f )  =  .0.  ) )
2118, 19, 20syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
( ( trL `  K
) `  W ) `  f ) ( le
`  K )  .0.  <->  ( ( ( trL `  K
) `  W ) `  f )  =  .0.  ) )
223, 4, 8, 13, 14trlid0b 33918 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( ( LTrn `  K
) `  W )
)  ->  ( f  =  (  _I  |`  B )  <-> 
( ( ( trL `  K ) `  W
) `  f )  =  .0.  ) )
2321, 22bitr4d 256 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
( ( trL `  K
) `  W ) `  f ) ( le
`  K )  .0.  <->  f  =  (  _I  |`  B ) ) )
2423rabbidva 2984 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f ) ( le
`  K )  .0. 
}  =  { f  e.  ( ( LTrn `  K ) `  W
)  |  f  =  (  _I  |`  B ) } )
253, 8, 13idltrn 33890 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
26 rabsn 3963 . . 3  |-  ( (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )  ->  { f  e.  ( ( LTrn `  K
) `  W )  |  f  =  (  _I  |`  B ) }  =  { (  _I  |`  B ) } )
2725, 26syl 16 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  { f  e.  ( ( LTrn `  K
) `  W )  |  f  =  (  _I  |`  B ) }  =  { (  _I  |`  B ) } )
2817, 24, 273eqtrd 2479 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { (  _I  |`  B ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2740   {csn 3898   class class class wbr 4313    _I cid 4652    |` cres 4863   ` cfv 5439   Basecbs 14195   lecple 14266   0.cp0 15228   AtLatcal 33005   HLchlt 33091   LHypclh 33724   LTrncltrn 33841   trLctrl 33898   DIsoAcdia 34769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-lhyp 33728  df-laut 33729  df-ldil 33844  df-ltrn 33845  df-trl 33899  df-disoa 34770
This theorem is referenced by:  dib0  34905
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