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Theorem cdleml3N 30434
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 30281 . . . . . 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 2482 . . . 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 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 30281 . . . . . 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 2482 . . . 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 30433 . . 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 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 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 30222 . . . . . 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 2292 . . . 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 30228 . . . . . . 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 30280 . . . . . 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 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 2656 . 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 936    = wceq 1624    e. wcel 1685    =/= wne 2447   E.wrex 2545    e. cmpt 4078    _I cid 4303    |` cres 4690    o. ccom 4692   ` cfv 5221   Basecbs 13142   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614   TEndoctendo 30208
This theorem is referenced by:  cdleml4N  30435
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-fal 1313  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tendo 30211
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