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Theorem dian0 31774
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 2435 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 30884 . . . 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 2435 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2435 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
81, 6, 2, 7trlid0 30910 . . . . 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 30095 . . . . . 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 30039 . . . . 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 4224 . . 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 31768 . . 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 3626 . 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 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   class class class wbr 4204    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   0.cp0 14458   AtLatcal 29999   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   DIsoAcdia 31763
This theorem is referenced by:  dialss  31781  dibn0  31888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-disoa 31764
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