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Theorem dian0 31155
Description: The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
dian0.b  |-  B  =  ( Base `  K
)
dian0.l  |-  .<_  =  ( le `  K )
dian0.h  |-  H  =  ( LHyp `  K
)
dian0.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
dian0  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )

Proof of Theorem dian0
StepHypRef Expression
1 dian0.b . . . . 5  |-  B  =  ( Base `  K
)
2 dian0.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2388 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 30265 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
54adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
6 eqid 2388 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2388 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
81, 6, 2, 7trlid0 30291 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  =  ( 0. `  K
) )
98adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( trL `  K
) `  W ) `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
10 hlatl 29476 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
1110adantr 452 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  AtLat )
12 simpl 444 . . . . 5  |-  ( ( X  e.  B  /\  X  .<_  W )  ->  X  e.  B )
13 dian0.l . . . . . 6  |-  .<_  =  ( le `  K )
141, 13, 6atl0le 29420 . . . . 5  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  ( 0. `  K )  .<_  X )
1511, 12, 14syl2an 464 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( 0. `  K )  .<_  X )
169, 15eqbrtrd 4174 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( trL `  K
) `  W ) `  (  _I  |`  B ) )  .<_  X )
17 dian0.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
181, 13, 2, 3, 7, 17diaelval 31149 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
(  _I  |`  B )  e.  ( I `  X )  <->  ( (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )  /\  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  .<_  X ) ) )
195, 16, 18mpbir2and 889 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (  _I  |`  B )  e.  ( I `  X
) )
20 ne0i 3578 . 2  |-  ( (  _I  |`  B )  e.  ( I `  X
)  ->  ( I `  X )  =/=  (/) )
2119, 20syl 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   (/)c0 3572   class class class wbr 4154    _I cid 4435    |` cres 4821   ` cfv 5395   Basecbs 13397   lecple 13464   0.cp0 14394   AtLatcal 29380   HLchlt 29466   LHypclh 30099   LTrncltrn 30216   trLctrl 30273   DIsoAcdia 31144
This theorem is referenced by:  dialss  31162  dibn0  31269
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274  df-disoa 31145
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