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Theorem dih1dimb2 35913
Description: Isomorphism H at an atom under  W. (Contributed by NM, 27-Apr-2014.)
Hypotheses
Ref Expression
dih1dimb2.b  |-  B  =  ( Base `  K
)
dih1dimb2.l  |-  .<_  =  ( le `  K )
dih1dimb2.a  |-  A  =  ( Atoms `  K )
dih1dimb2.h  |-  H  =  ( LHyp `  K
)
dih1dimb2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dih1dimb2.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
dih1dimb2.u  |-  U  =  ( ( DVecH `  K
) `  W )
dih1dimb2.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dih1dimb2.n  |-  N  =  ( LSpan `  U )
Assertion
Ref Expression
dih1dimb2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
Distinct variable groups:    .<_ , f    A, f    B, h    f, H   
f, h, K    Q, f    T, f, h    f, W, h
Allowed substitution hints:    A( h)    B( f)    Q( h)    U( f, h)    H( h)    I( f, h)   
.<_ ( h)    N( f, h)    O( f, h)

Proof of Theorem dih1dimb2
StepHypRef Expression
1 dih1dimb2.l . . 3  |-  .<_  =  ( le `  K )
2 dih1dimb2.a . . 3  |-  A  =  ( Atoms `  K )
3 dih1dimb2.h . . 3  |-  H  =  ( LHyp `  K
)
4 dih1dimb2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
5 eqid 2460 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
61, 2, 3, 4, 5cdlemf 35234 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  E. f  e.  T  ( (
( trL `  K
) `  W ) `  f )  =  Q )
7 simp3 993 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
( ( trL `  K
) `  W ) `  f )  =  Q )
8 simp1rl 1056 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  Q  e.  A )
97, 8eqeltrd 2548 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
( ( trL `  K
) `  W ) `  f )  e.  A
)
10 simp1l 1015 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simp2 992 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  f  e.  T )
12 dih1dimb2.b . . . . . . . 8  |-  B  =  ( Base `  K
)
1312, 2, 3, 4, 5trlnidatb 34848 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( f  =/=  (  _I  |`  B )  <-> 
( ( ( trL `  K ) `  W
) `  f )  e.  A ) )
1410, 11, 13syl2anc 661 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
f  =/=  (  _I  |`  B )  <->  ( (
( trL `  K
) `  W ) `  f )  e.  A
) )
159, 14mpbird 232 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  f  =/=  (  _I  |`  B ) )
167fveq2d 5861 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
I `  ( (
( trL `  K
) `  W ) `  f ) )  =  ( I `  Q
) )
17 dih1dimb2.o . . . . . . . 8  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
18 dih1dimb2.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
19 dih1dimb2.i . . . . . . . 8  |-  I  =  ( ( DIsoH `  K
) `  W )
20 dih1dimb2.n . . . . . . . 8  |-  N  =  ( LSpan `  U )
2112, 3, 4, 5, 17, 18, 19, 20dih1dimb 35912 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( I `  ( ( ( trL `  K ) `  W
) `  f )
)  =  ( N `
 { <. f ,  O >. } ) )
2210, 11, 21syl2anc 661 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
I `  ( (
( trL `  K
) `  W ) `  f ) )  =  ( N `  { <. f ,  O >. } ) )
2316, 22eqtr3d 2503 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
I `  Q )  =  ( N `  { <. f ,  O >. } ) )
2415, 23jca 532 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T  /\  ( ( ( trL `  K ) `  W
) `  f )  =  Q )  ->  (
f  =/=  (  _I  |`  B )  /\  (
I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
25243expia 1193 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  /\  f  e.  T
)  ->  ( (
( ( trL `  K
) `  W ) `  f )  =  Q  ->  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) ) )
2625reximdva 2931 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  ( E. f  e.  T  ( ( ( trL `  K ) `  W
) `  f )  =  Q  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) ) )
276, 26mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   {csn 4020   <.cop 4026   class class class wbr 4440    |-> cmpt 4498    _I cid 4783    |` cres 4994   ` cfv 5579   Basecbs 14479   lecple 14551   LSpanclspn 17393   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829   DVecHcdvh 35750   DIsoHcdih 35900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-undef 6992  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-0g 14686  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-drng 17174  df-lmod 17290  df-lss 17355  df-lsp 17394  df-lvec 17525  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830  df-tendo 35426  df-edring 35428  df-disoa 35701  df-dvech 35751  df-dib 35811  df-dih 35901
This theorem is referenced by:  dihatlat  36006  dihatexv  36010
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