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Theorem cdleml3N 30985
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 955 . . 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 956 . . 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 991 . . 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 992 . . . 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 988 . . . . . 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 990 . . . . . 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 30832 . . . . . 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 1186 . . . . 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 2514 . . . 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 223 . . 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 993 . . . 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 989 . . . . . 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 30832 . . . . . 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 1186 . . . . 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 2514 . . . 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 223 . . 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 30984 . . 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 1194 . 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 958 . . . . . 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 447 . . . . . 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 1033 . . . . . 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 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 )  ->  f  e.  T )
298, 9, 10tendocoval 30773 . . . . . 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 1187 . . . . 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 2324 . . . 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 985 . . . . . 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 956 . . . . . . 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 1087 . . . . . . 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 30779 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  U  e.  E
)  ->  ( s  o.  U )  e.  E
)
3632, 33, 34, 35syl3anc 1182 . . . . . 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 1088 . . . . . 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 957 . . . . . 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 1089 . . . . . 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 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 ) )  ->  f  =/=  (  _I  |`  B ) )
417, 8, 9, 10tendocan 30831 . . . . . 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 1200 . . . . 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 1153 . . . 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 226 . . 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 2689 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578    e. cmpt 4114    _I cid 4341    |` cres 4728    o. ccom 4730   ` cfv 5292   Basecbs 13195   HLchlt 29358   LHypclh 29991   LTrncltrn 30108   trLctrl 30165   TEndoctendo 30759
This theorem is referenced by:  cdleml4N  30986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-map 6817  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507  df-lines 29508  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166  df-tendo 30762
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