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Theorem cdleml4N 35650
Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleml1.b  |-  B  =  ( Base `  K
)
cdleml1.h  |-  H  =  ( LHyp `  K
)
cdleml1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdleml1.r  |-  R  =  ( ( trL `  K
) `  W )
cdleml1.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdleml3.o  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
cdleml4N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  ) )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
Distinct variable groups:    E, s    K, s    R, s    T, s    U, s    V, s    W, s, g    B, g, s   
g, H, s    g, K    .0. , s    T, g    g, W
Allowed substitution hints:    R( g)    U( g)    E( g)    V( g)    .0. ( g)

Proof of Theorem cdleml4N
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cdleml1.b . . . 4  |-  B  =  ( Base `  K
)
2 cdleml1.h . . . 4  |-  H  =  ( LHyp `  K
)
3 cdleml1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 35239 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. f  e.  T  f  =/=  (  _I  |`  B ) )
543ad2ant1 1012 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  ) )  ->  E. f  e.  T  f  =/=  (  _I  |`  B ) )
6 simp11 1021 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simp12l 1104 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  U  e.  E
)
8 simp12r 1105 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  V  e.  E
)
9 simp2 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  f  e.  T
)
10 simp3 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  f  =/=  (  _I  |`  B ) )
11 simp13l 1106 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  U  =/=  .0.  )
12 simp13r 1107 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  V  =/=  .0.  )
13 cdleml1.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
14 cdleml1.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
15 cdleml3.o . . . . 5  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
161, 2, 3, 13, 14, 15cdleml3N 35649 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
176, 7, 8, 9, 10, 11, 12, 16syl133anc 1246 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  )
)  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
1817rexlimdv3a 2950 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  ) )  -> 
( E. f  e.  T  f  =/=  (  _I  |`  B )  ->  E. s  e.  E  ( s  o.  U
)  =  V ) )
195, 18mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  ) )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808    |-> cmpt 4498    _I cid 4783    |` cres 4994    o. ccom 4996   ` cfv 5579   Basecbs 14479   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829   TEndoctendo 35423
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-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-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  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-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  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-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
This theorem is referenced by:  cdleml5N  35651
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