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Theorem dicn0 34192
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 455 . . . . . 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 2402 . . . . . . . 8  |-  ( oc
`  K )  =  ( oc `  K
)
4 dicn0.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 dicn0.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
62, 3, 4, 5lhpocnel 33015 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
76adantr 463 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
8 simpr 459 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
9 eqid 2402 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2402 . . . . . . 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 33578 . . . . . 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 1230 . . . . 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 2402 . . . . . 6  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
14 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1513, 14tendo02 33786 . . . . 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 17 . . . 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 2410 . . 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 2402 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
1914, 5, 9, 18, 13tendo0cl 33789 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) )  e.  ( (
TEndo `  K ) `  W ) )
2019adantr 463 . . 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 2402 . . . 4  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
22 dicn0.i . . . 4  |-  I  =  ( ( DIsoC `  K
) `  W )
23 fvex 5858 . . . . 5  |-  ( Base `  K )  e.  _V
24 resiexg 6719 . . . . 5  |-  ( (
Base `  K )  e.  _V  ->  (  _I  |`  ( Base `  K
) )  e.  _V )
2523, 24ax-mp 5 . . . 4  |-  (  _I  |`  ( Base `  K
) )  e.  _V
26 fvex 5858 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
2726mptex 6123 . . . 4  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  e.  _V
282, 4, 5, 21, 9, 18, 22, 25, 27dicopelval 34177 . . 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 923 . 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 3743 . 2  |-  ( <.
(  _I  |`  ( Base `  K ) ) ,  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) >.  e.  (
I `  Q )  ->  ( I `  Q
)  =/=  (/) )
3129, 30syl 17 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3058   (/)c0 3737   <.cop 3977   class class class wbr 4394    |-> cmpt 4452    _I cid 4732    |` cres 4824   ` cfv 5568   iota_crio 6238   Basecbs 14839   lecple 14914   occoc 14915   Atomscatm 32261   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   TEndoctendo 33751   DIsoCcdic 34172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-tendo 33754  df-dic 34173
This theorem is referenced by:  diclss  34193
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