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Theorem dian0 31851
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 2296 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 30961 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
54adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
6 eqid 2296 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2296 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
81, 6, 2, 7trlid0 30987 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  =  ( 0. `  K
) )
98adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( trL `  K
) `  W ) `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
10 hlatl 30172 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
1110adantr 451 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  AtLat )
12 simpl 443 . . . . 5  |-  ( ( X  e.  B  /\  X  .<_  W )  ->  X  e.  B )
13 dian0.l . . . . . 6  |-  .<_  =  ( le `  K )
141, 13, 6atl0le 30116 . . . . 5  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  ( 0. `  K )  .<_  X )
1511, 12, 14syl2an 463 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( 0. `  K )  .<_  X )
169, 15eqbrtrd 4059 . . 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 31845 . . 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 888 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (  _I  |`  B )  e.  ( I `  X
) )
20 ne0i 3474 . 2  |-  ( (  _I  |`  B )  e.  ( I `  X
)  ->  ( I `  X )  =/=  (/) )
2119, 20syl 15 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   class class class wbr 4039    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   AtLatcal 30076   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   DIsoAcdia 31840
This theorem is referenced by:  dialss  31858  dibn0  31965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-disoa 31841
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