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Theorem cdleml3N 31460
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 957 . . 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 958 . . 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 993 . . 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 994 . . . 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 990 . . . . . 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 992 . . . . . 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 31307 . . . . . 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 1188 . . . . 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 2602 . . . 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 224 . . 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 995 . . . 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 991 . . . . . 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 31307 . . . . . 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 1188 . . . . 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 2602 . . . 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 224 . . 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 31459 . . 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 1196 . 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 960 . . . . . 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 448 . . . . . 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 1035 . . . . . 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 1037 . . . . . 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 31248 . . . . . 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 1189 . . . . 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 2412 . . . 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 987 . . . . . 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 958 . . . . . . 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 1089 . . . . . . 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 31254 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  U  e.  E
)  ->  ( s  o.  U )  e.  E
)
3632, 33, 34, 35syl3anc 1184 . . . . . 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 1090 . . . . . 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 959 . . . . . 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 1091 . . . . . 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 1092 . . . . . 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 31306 . . . . . 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 1202 . . . . 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 1155 . . . 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 227 . . 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 2778 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    e. cmpt 4226    _I cid 4453    |` cres 4839    o. ccom 4841   ` cfv 5413   Basecbs 13424   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640   TEndoctendo 31234
This theorem is referenced by:  cdleml4N  31461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237
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