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Theorem cdleml3N 29856
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 960 . . 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 961 . . 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 996 . . 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 997 . . . 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 993 . . . . . 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 995 . . . . . 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 29703 . . . . . 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 1191 . . . . 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 2447 . . . 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 225 . . 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 998 . . . 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 994 . . . . . 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 29703 . . . . . 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 1191 . . . . 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 2447 . . . 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 225 . . 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 29855 . . 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 1199 . 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 963 . . . . . 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 449 . . . . . 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 1038 . . . . . 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 1040 . . . . . 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 29644 . . . . . 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 1192 . . . . 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 2261 . . . 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 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 ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
33 simp2 961 . . . . . . 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 1092 . . . . . . 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 29650 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  U  e.  E
)  ->  ( s  o.  U )  e.  E
)
3632, 33, 34, 35syl3anc 1187 . . . . . 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 1093 . . . . . 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 962 . . . . . 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 1094 . . . . . 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 1095 . . . . . 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 29702 . . . . . 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 1205 . . . . 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 1158 . . . 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 228 . . 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 2617 . 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 16 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510    e. cmpt 3974    _I cid 4197    |` cres 4582    o. ccom 4584   ` cfv 4592   Basecbs 13022   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036   TEndoctendo 29630
This theorem is referenced by:  cdleml4N  29857
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037  df-tendo 29633
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