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Theorem cdleml7 31953
Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
Hypotheses
Ref Expression
cdleml6.b  |-  B  =  ( Base `  K
)
cdleml6.j  |-  .\/  =  ( join `  K )
cdleml6.m  |-  ./\  =  ( meet `  K )
cdleml6.h  |-  H  =  ( LHyp `  K
)
cdleml6.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdleml6.r  |-  R  =  ( ( trL `  K
) `  W )
cdleml6.p  |-  Q  =  ( ( oc `  K ) `  W
)
cdleml6.z  |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( (
h `  Q )  .\/  ( R `  (
b  o.  `' ( s `  h ) ) ) ) )
cdleml6.y  |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdleml6.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  -> 
( z `  Q
)  =  Y ) )
cdleml6.u  |-  U  =  ( g  e.  T  |->  if ( ( s `
 h )  =  h ,  g ,  X ) )
cdleml6.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdleml6.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
cdleml7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  (
( U  o.  s
) `  h )  =  ( (  _I  |`  T ) `  h
) )
Distinct variable groups:    g, b,
z,  ./\    .\/ , b, g, z    B, b, f, g, z   
h, b, g, z   
s, b, g, z    H, b, g, z    K, b, g, z    Q, b, g, z    R, b, g, z    T, b, f, g, z    W, b, g, z    z, Y   
g, Z
Allowed substitution hints:    B( h, s)    Q( f, h, s)    R( f, h, s)    T( h, s)    U( z, f, g, h, s, b)    E( z, f, g, h, s, b)    H( f, h, s)    .\/ ( f, h, s)    K( f, h, s)    ./\ ( f, h, s)    W( f, h, s)    X( z, f, g, h, s, b)    Y( f, g, h, s, b)    .0. ( z, f, g, h, s, b)    Z( z, f, h, s, b)

Proof of Theorem cdleml7
StepHypRef Expression
1 cdleml6.b . . . 4  |-  B  =  ( Base `  K
)
2 cdleml6.j . . . 4  |-  .\/  =  ( join `  K )
3 cdleml6.m . . . 4  |-  ./\  =  ( meet `  K )
4 cdleml6.h . . . 4  |-  H  =  ( LHyp `  K
)
5 cdleml6.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
6 cdleml6.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
7 cdleml6.p . . . 4  |-  Q  =  ( ( oc `  K ) `  W
)
8 cdleml6.z . . . 4  |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( (
h `  Q )  .\/  ( R `  (
b  o.  `' ( s `  h ) ) ) ) )
9 cdleml6.y . . . 4  |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
10 cdleml6.x . . . 4  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  -> 
( z `  Q
)  =  Y ) )
11 cdleml6.u . . . 4  |-  U  =  ( g  e.  T  |->  if ( ( s `
 h )  =  h ,  g ,  X ) )
12 cdleml6.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
13 cdleml6.o . . . 4  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13cdleml6 31952 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  e.  E  /\  ( U `  ( s `
 h ) )  =  h ) )
1514simprd 451 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U `  ( s `  h ) )  =  h )
16 simp1 958 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
1714simpld 447 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  e.  E )
18 simp3l 986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  s  e.  E )
19 simp2 959 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  h  e.  T )
204, 5, 12tendocoval 31737 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  s  e.  E )  /\  h  e.  T )  ->  (
( U  o.  s
) `  h )  =  ( U `  ( s `  h
) ) )
2116, 17, 18, 19, 20syl121anc 1190 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  (
( U  o.  s
) `  h )  =  ( U `  ( s `  h
) ) )
22 fvresi 5960 . . 3  |-  ( h  e.  T  ->  (
(  _I  |`  T ) `
 h )  =  h )
23223ad2ant2 980 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  (
(  _I  |`  T ) `
 h )  =  h )
2415, 21, 233eqtr4d 2485 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  (
( U  o.  s
) `  h )  =  ( (  _I  |`  T ) `  h
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   A.wral 2712   ifcif 3767    e. cmpt 4297    _I cid 4528   `'ccnv 4912    |` cres 4915    o. ccom 4917   ` cfv 5489  (class class class)co 6117   iota_crio 6578   Basecbs 13507   occoc 13575   joincjn 14439   meetcmee 14440   HLchlt 30322   LHypclh 30955   LTrncltrn 31072   trLctrl 31129   TEndoctendo 31723
This theorem is referenced by:  cdleml8  31954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-iin 4125  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-map 7056  df-poset 14441  df-plt 14453  df-lub 14469  df-glb 14470  df-join 14471  df-meet 14472  df-p0 14506  df-p1 14507  df-lat 14513  df-clat 14575  df-oposet 30148  df-ol 30150  df-oml 30151  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-llines 30469  df-lplanes 30470  df-lvols 30471  df-lines 30472  df-psubsp 30474  df-pmap 30475  df-padd 30767  df-lhyp 30959  df-laut 30960  df-ldil 31075  df-ltrn 31076  df-trl 31130  df-tendo 31726
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