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Theorem cdleml6 31509
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
cdleml6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  e.  E  /\  ( U `  ( s `
 h ) )  =  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 cdleml6
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp3l 985 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  s  e.  E )
3 simp2 958 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  h  e.  T )
4 cdleml6.h . . . 4  |-  H  =  ( LHyp `  K
)
5 cdleml6.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
6 cdleml6.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
74, 5, 6tendocl 31295 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  h  e.  T
)  ->  ( s `  h )  e.  T
)
81, 2, 3, 7syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  (
s `  h )  e.  T )
9 cdleml6.b . . . 4  |-  B  =  ( Base `  K
)
10 cdleml6.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
11 cdleml6.o . . . 4  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
129, 4, 5, 10, 6, 11tendotr 31358 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  s  =/= 
.0.  )  /\  h  e.  T )  ->  ( R `  ( s `  h ) )  =  ( R `  h
) )
13123com23 1159 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( R `  ( s `  h ) )  =  ( R `  h
) )
14 cdleml6.j . . 3  |-  .\/  =  ( join `  K )
15 cdleml6.m . . 3  |-  ./\  =  ( meet `  K )
16 eqid 2430 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
17 eqid 2430 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
18 cdleml6.p . . 3  |-  Q  =  ( ( oc `  K ) `  W
)
19 cdleml6.z . . 3  |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( (
h `  Q )  .\/  ( R `  (
b  o.  `' ( s `  h ) ) ) ) )
20 cdleml6.y . . 3  |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
21 cdleml6.x . . 3  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  -> 
( z `  Q
)  =  Y ) )
22 cdleml6.u . . 3  |-  U  =  ( g  e.  T  |->  if ( ( s `
 h )  =  h ,  g ,  X ) )
239, 14, 15, 16, 17, 4, 5, 10, 18, 19, 20, 21, 22, 6cdlemk56w 31501 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s `
 h )  e.  T  /\  h  e.  T )  /\  ( R `  ( s `  h ) )  =  ( R `  h
) )  ->  ( U  e.  E  /\  ( U `  ( s `
 h ) )  =  h ) )
241, 8, 3, 13, 23syl121anc 1189 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  e.  E  /\  ( U `  ( s `
 h ) )  =  h ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   A.wral 2692   ifcif 3726    e. cmpt 4253    _I cid 4480   `'ccnv 4863    |` cres 4866    o. ccom 4868   ` cfv 5440  (class class class)co 6067   iota_crio 6528   Basecbs 13452   occoc 13520   joincjn 14384   meetcmee 14385   Atomscatm 29792   HLchlt 29879   LHypclh 30512   LTrncltrn 30629   trLctrl 30686   TEndoctendo 31280
This theorem is referenced by:  cdleml7  31510  cdleml8  31511  erngdvlem4  31519  erngdvlem4-rN  31527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-map 7006  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028  df-lines 30029  df-psubsp 30031  df-pmap 30032  df-padd 30324  df-lhyp 30516  df-laut 30517  df-ldil 30632  df-ltrn 30633  df-trl 30687  df-tendo 31283
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