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Theorem cdlemfnid 30805
Description: cdlemf 30804 with additional constraint of non-identity. (Contributed by NM, 20-Jun-2013.)
Hypotheses
Ref Expression
cdlemfnid.b  |-  B  =  ( Base `  K
)
cdlemfnid.l  |-  .<_  =  ( le `  K )
cdlemfnid.a  |-  A  =  ( Atoms `  K )
cdlemfnid.h  |-  H  =  ( LHyp `  K
)
cdlemfnid.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemfnid.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemfnid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. f  e.  T  ( ( R `  f )  =  U  /\  f  =/=  (  _I  |`  B ) ) )
Distinct variable groups:    A, f    f, H    f, K    .<_ , f    T, f    U, f    f, W
Allowed substitution hints:    B( f)    R( f)

Proof of Theorem cdlemfnid
StepHypRef Expression
1 cdlemfnid.l . . 3  |-  .<_  =  ( le `  K )
2 cdlemfnid.a . . 3  |-  A  =  ( Atoms `  K )
3 cdlemfnid.h . . 3  |-  H  =  ( LHyp `  K
)
4 cdlemfnid.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
5 cdlemfnid.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
61, 2, 3, 4, 5cdlemf 30804 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. f  e.  T  ( R `  f )  =  U )
7 simp3 957 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  ( R `  f )  =  U )
8 simp1rl 1020 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  U  e.  A
)
97, 8eqeltrd 2432 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  ( R `  f )  e.  A
)
10 simp1l 979 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simp2 956 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  f  e.  T
)
12 cdlemfnid.b . . . . . . . 8  |-  B  =  ( Base `  K
)
1312, 2, 3, 4, 5trlnidatb 30418 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( f  =/=  (  _I  |`  B )  <-> 
( R `  f
)  e.  A ) )
1410, 11, 13syl2anc 642 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  ( f  =/=  (  _I  |`  B )  <-> 
( R `  f
)  e.  A ) )
159, 14mpbird 223 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  f  =/=  (  _I  |`  B ) )
167, 15jca 518 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T  /\  ( R `  f
)  =  U )  ->  ( ( R `
 f )  =  U  /\  f  =/=  (  _I  |`  B ) ) )
17163expia 1153 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  /\  f  e.  T
)  ->  ( ( R `  f )  =  U  ->  ( ( R `  f )  =  U  /\  f  =/=  (  _I  |`  B ) ) ) )
1817reximdva 2731 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( E. f  e.  T  ( R `  f )  =  U  ->  E. f  e.  T  ( ( R `  f )  =  U  /\  f  =/=  (  _I  |`  B ) ) ) )
196, 18mpd 14 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  E. f  e.  T  ( ( R `  f )  =  U  /\  f  =/=  (  _I  |`  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   class class class wbr 4102    _I cid 4383    |` cres 4770   ` cfv 5334   Basecbs 13239   lecple 13306   Atomscatm 29505   HLchlt 29592   LHypclh 30225   LTrncltrn 30342   trLctrl 30399
This theorem is referenced by:  cdlemftr3  30806  cdlemj3  31064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-map 6859  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-p1 14239  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-llines 29739  df-lplanes 29740  df-lvols 29741  df-lines 29742  df-psubsp 29744  df-pmap 29745  df-padd 30037  df-lhyp 30229  df-laut 30230  df-ldil 30345  df-ltrn 30346  df-trl 30400
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