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Theorem cdleml3N 34930
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
cdleml3N  |-  ( ( ( 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 )
Distinct variable groups:    E, s    K, s    R, s    T, s    U, s    V, s    W, s, f, g    B, g, s    f, E    f,
g, H, s    f, K, g    .0. , f, s    T, f, g    U, f   
f, V    f, W, g
Allowed substitution hints:    B( f)    R( f, g)    U( g)    E( g)    V( g)    .0. ( g)

Proof of Theorem cdleml3N
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp2 989 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( U  e.  E  /\  V  e.  E  /\  f  e.  T
) )
3 simp31 1024 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
f  =/=  (  _I  |`  B ) )
4 simp32 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  ->  U  =/=  .0.  )
5 simp21 1021 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  ->  U  e.  E )
6 simp23 1023 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
f  e.  T )
7 cdleml1.b . . . . . . 7  |-  B  =  ( Base `  K
)
8 cdleml1.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
9 cdleml1.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
10 cdleml1.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
11 cdleml3.o . . . . . . 7  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
127, 8, 9, 10, 11tendoid0 34777 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( f  e.  T  /\  f  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  f )  =  (  _I  |`  B )  <-> 
U  =  .0.  )
)
131, 5, 6, 3, 12syl112anc 1223 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( ( U `  f )  =  (  _I  |`  B )  <->  U  =  .0.  ) )
1413necon3bid 2706 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( ( U `  f )  =/=  (  _I  |`  B )  <->  U  =/=  .0.  ) )
154, 14mpbird 232 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( U `  f
)  =/=  (  _I  |`  B ) )
16 simp33 1026 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  ->  V  =/=  .0.  )
17 simp22 1022 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  ->  V  e.  E )
187, 8, 9, 10, 11tendoid0 34777 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  ( f  e.  T  /\  f  =/=  (  _I  |`  B ) ) )  ->  ( ( V `  f )  =  (  _I  |`  B )  <-> 
V  =  .0.  )
)
191, 17, 6, 3, 18syl112anc 1223 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( ( V `  f )  =  (  _I  |`  B )  <->  V  =  .0.  ) )
2019necon3bid 2706 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( ( V `  f )  =/=  (  _I  |`  B )  <->  V  =/=  .0.  ) )
2116, 20mpbird 232 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/= 
.0.  ) )  -> 
( V `  f
)  =/=  (  _I  |`  B ) )
22 cdleml1.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
237, 8, 9, 22, 10cdleml2N 34929 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  E. s  e.  E  ( s `  ( U `  f
) )  =  ( V `  f ) )
241, 2, 3, 15, 21, 23syl113anc 1231 . 2  |-  ( ( ( 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 `  ( U `  f )
)  =  ( V `
 f ) )
25 simpl1 991 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  ( K  e.  HL  /\  W  e.  H ) )
26 simpr 461 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  s  e.  E )
27 simpl21 1066 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  U  e.  E )
28 simpl23 1068 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  f  e.  T )
298, 9, 10tendocoval 34718 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  U  e.  E )  /\  f  e.  T )  ->  (
( s  o.  U
) `  f )  =  ( s `  ( U `  f ) ) )
3025, 26, 27, 28, 29syl121anc 1224 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  ( ( s  o.  U ) `  f
)  =  ( s `
 ( U `  f ) ) )
3130eqeq1d 2453 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  ( ( ( s  o.  U ) `  f )  =  ( V `  f )  <-> 
( s `  ( U `  f )
)  =  ( V `
 f ) ) )
32 simp11 1018 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
33 simp2 989 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  s  e.  E
)
34 simp121 1120 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  U  e.  E
)
358, 10tendococl 34724 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  U  e.  E
)  ->  ( s  o.  U )  e.  E
)
3632, 33, 34, 35syl3anc 1219 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  ( s  o.  U )  e.  E
)
37 simp122 1121 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  V  e.  E
)
38 simp3 990 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  ( ( s  o.  U ) `  f )  =  ( V `  f ) )
39 simp123 1122 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  f  e.  T
)
40 simp131 1123 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  f  =/=  (  _I  |`  B ) )
417, 8, 9, 10tendocan 34776 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s  o.  U )  e.  E  /\  V  e.  E  /\  ( ( s  o.  U ) `
 f )  =  ( V `  f
) )  /\  (
f  e.  T  /\  f  =/=  (  _I  |`  B ) ) )  ->  (
s  o.  U )  =  V )
4232, 36, 37, 38, 39, 40, 41syl132anc 1237 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E  /\  ( ( s  o.  U ) `  f
)  =  ( V `
 f ) )  ->  ( s  o.  U )  =  V )
43423expia 1190 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  ( ( ( s  o.  U ) `  f )  =  ( V `  f )  ->  ( s  o.  U )  =  V ) )
4431, 43sylbird 235 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  ( f  =/=  (  _I  |`  B )  /\  U  =/=  .0.  /\  V  =/=  .0.  ) )  /\  s  e.  E )  ->  ( ( s `  ( U `  f ) )  =  ( V `
 f )  -> 
( s  o.  U
)  =  V ) )
4544reximdva 2926 . 2  |-  ( ( ( 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 `  ( U `  f ) )  =  ( V `
 f )  ->  E. s  e.  E  ( s  o.  U
)  =  V ) )
4624, 45mpd 15 1  |-  ( ( ( 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 )
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    |-> cmpt 4450    _I cid 4731    |` cres 4942    o. ccom 4944   ` cfv 5518   Basecbs 14278   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   trLctrl 34110   TEndoctendo 34704
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-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-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  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-1st 6679  df-2nd 6680  df-undef 6894  df-map 7318  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-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
This theorem is referenced by:  cdleml4N  34931
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