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Theorem dih1dimb2 35194
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 2451 . . 3  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
61, 2, 3, 4, 5cdlemf 34515 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  E. f  e.  T  ( (
( trL `  K
) `  W ) `  f )  =  Q )
7 simp3 990 . . . . . . 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 1053 . . . . . . 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 2539 . . . . . 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 1012 . . . . . . 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 989 . . . . . . 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 34129 . . . . . . 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 5795 . . . . . 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 35193 . . . . . . 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 2494 . . . . 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 1190 . . 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 2926 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   {csn 3977   <.cop 3983   class class class wbr 4392    |-> cmpt 4450    _I cid 4731    |` cres 4942   ` cfv 5518   Basecbs 14278   lecple 14349   LSpanclspn 17160   Atomscatm 33216   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   trLctrl 34110   DVecHcdvh 35031   DIsoHcdih 35181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-tpos 6847  df-undef 6894  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-0g 14484  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-mnd 15519  df-grp 15649  df-minusg 15650  df-sbg 15651  df-mgp 16699  df-ur 16711  df-rng 16755  df-oppr 16823  df-dvdsr 16841  df-unit 16842  df-invr 16872  df-dvr 16883  df-drng 16942  df-lmod 17058  df-lss 17122  df-lsp 17161  df-lvec 17292  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111  df-tendo 34707  df-edring 34709  df-disoa 34982  df-dvech 35032  df-dib 35092  df-dih 35182
This theorem is referenced by:  dihatlat  35287  dihatexv  35291
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