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Theorem dicn0 31991
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 445 . . . . . 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 2437 . . . . . . . 8  |-  ( oc
`  K )  =  ( oc `  K
)
4 dicn0.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 dicn0.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
62, 3, 4, 5lhpocnel 30816 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
76adantr 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
8 simpr 449 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
9 eqid 2437 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2437 . . . . . . 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 31377 . . . . . 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 1185 . . . . 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 2437 . . . . . 6  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
14 eqid 2437 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1513, 14tendo02 31585 . . . . 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 2442 . . 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 2437 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
1914, 5, 9, 18, 13tendo0cl 31588 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) )  e.  ( (
TEndo `  K ) `  W ) )
2019adantr 453 . . 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 2437 . . . 4  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
22 dicn0.i . . . 4  |-  I  =  ( ( DIsoC `  K
) `  W )
23 fvex 5743 . . . . 5  |-  ( Base `  K )  e.  _V
24 resiexg 5189 . . . . 5  |-  ( (
Base `  K )  e.  _V  ->  (  _I  |`  ( Base `  K
) )  e.  _V )
2523, 24ax-mp 8 . . . 4  |-  (  _I  |`  ( Base `  K
) )  e.  _V
26 fvex 5743 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
2726mptex 5967 . . . 4  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  e.  _V
282, 4, 5, 21, 9, 18, 22, 25, 27dicopelval 31976 . . 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 890 . 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 3635 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   _Vcvv 2957   (/)c0 3629   <.cop 3818   class class class wbr 4213    e. cmpt 4267    _I cid 4494    |` cres 4881   ` cfv 5455   iota_crio 6543   Basecbs 13470   lecple 13537   occoc 13538   Atomscatm 30062   HLchlt 30149   LHypclh 30782   LTrncltrn 30899   TEndoctendo 31550   DIsoCcdic 31971
This theorem is referenced by:  diclss  31992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-map 7021  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297  df-lvols 30298  df-lines 30299  df-psubsp 30301  df-pmap 30302  df-padd 30594  df-lhyp 30786  df-laut 30787  df-ldil 30902  df-ltrn 30903  df-trl 30957  df-tendo 31553  df-dic 31972
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