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Theorem dicn0 35989
Description: The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)
Hypotheses
Ref Expression
dicn0.l  |-  .<_  =  ( le `  K )
dicn0.a  |-  A  =  ( Atoms `  K )
dicn0.h  |-  H  =  ( LHyp `  K
)
dicn0.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicn0  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =/=  (/) )

Proof of Theorem dicn0
Dummy variables  g 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 dicn0.l . . . . . . . 8  |-  .<_  =  ( le `  K )
3 eqid 2467 . . . . . . . 8  |-  ( oc
`  K )  =  ( oc `  K
)
4 dicn0.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 dicn0.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
62, 3, 4, 5lhpocnel 34814 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
76adantr 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
8 simpr 461 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
9 eqid 2467 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2467 . . . . . . 7  |-  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  =  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )
112, 4, 5, 9, 10ltrniotacl 35375 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  e.  ( (
LTrn `  K ) `  W ) )
121, 7, 8, 11syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q )  e.  ( ( LTrn `  K ) `  W
) )
13 eqid 2467 . . . . . 6  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
14 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1513, 14tendo02 35583 . . . . 5  |-  ( (
iota_ g  e.  (
( LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q )  e.  ( ( LTrn `  K
) `  W )  ->  ( ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) )  =  (  _I  |`  ( Base `  K ) ) )
1612, 15syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) )  =  (  _I  |`  ( Base `  K ) ) )
1716eqcomd 2475 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
(  _I  |`  ( Base `  K ) )  =  ( ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) ) )
18 eqid 2467 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
1914, 5, 9, 18, 13tendo0cl 35586 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) )  e.  ( (
TEndo `  K ) `  W ) )
2019adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) )  e.  ( (
TEndo `  K ) `  W ) )
21 eqid 2467 . . . 4  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
22 dicn0.i . . . 4  |-  I  =  ( ( DIsoC `  K
) `  W )
23 fvex 5874 . . . . 5  |-  ( Base `  K )  e.  _V
24 resiexg 6717 . . . . 5  |-  ( (
Base `  K )  e.  _V  ->  (  _I  |`  ( Base `  K
) )  e.  _V )
2523, 24ax-mp 5 . . . 4  |-  (  _I  |`  ( Base `  K
) )  e.  _V
26 fvex 5874 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
2726mptex 6129 . . . 4  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  e.  _V
282, 4, 5, 21, 9, 18, 22, 25, 27dicopelval 35974 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. (  _I  |`  ( Base `  K ) ) ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) >.  e.  (
I `  Q )  <->  ( (  _I  |`  ( Base `  K ) )  =  ( ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) )  /\  (
f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) )  e.  ( ( TEndo `  K
) `  W )
) ) )
2917, 20, 28mpbir2and 920 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  <. (  _I  |`  ( Base `  K ) ) ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) >.  e.  (
I `  Q )
)
30 ne0i 3791 . 2  |-  ( <.
(  _I  |`  ( Base `  K ) ) ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) >.  e.  (
I `  Q )  ->  ( I `  Q
)  =/=  (/) )
3129, 30syl 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    |` cres 5001   ` cfv 5586   iota_crio 6242   Basecbs 14486   lecple 14558   occoc 14559   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   DIsoCcdic 35969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tendo 35551  df-dic 35970
This theorem is referenced by:  diclss  35990
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