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Theorem cdleml8 31231
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
cdleml8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  =  (  _I  |`  T ) )
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 cdleml8
StepHypRef Expression
1 simp1 956 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 cdleml6.b . . . . . 6  |-  B  =  ( Base `  K
)
3 cdleml6.j . . . . . 6  |-  .\/  =  ( join `  K )
4 cdleml6.m . . . . . 6  |-  ./\  =  ( meet `  K )
5 cdleml6.h . . . . . 6  |-  H  =  ( LHyp `  K
)
6 cdleml6.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
7 cdleml6.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
8 cdleml6.p . . . . . 6  |-  Q  =  ( ( oc `  K ) `  W
)
9 cdleml6.z . . . . . 6  |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( (
h `  Q )  .\/  ( R `  (
b  o.  `' ( s `  h ) ) ) ) )
10 cdleml6.y . . . . . 6  |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
11 cdleml6.x . . . . . 6  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  -> 
( z `  Q
)  =  Y ) )
12 cdleml6.u . . . . . 6  |-  U  =  ( g  e.  T  |->  if ( ( s `
 h )  =  h ,  g ,  X ) )
13 cdleml6.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
14 cdleml6.o . . . . . 6  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cdleml6 31229 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  e.  E  /\  ( U `  ( s `
 h ) )  =  h ) )
16153adant2r 1178 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  e.  E  /\  ( U `  (
s `  h )
)  =  h ) )
1716simpld 445 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  e.  E )
18 simp3l 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
s  e.  E )
195, 13tendococl 31020 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  s  e.  E
)  ->  ( U  o.  s )  e.  E
)
201, 17, 18, 19syl3anc 1183 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  e.  E )
215, 6, 13tendoidcl 31017 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
22213ad2ant1 977 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
(  _I  |`  T )  e.  E )
232, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cdleml7 31230 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  (
( U  o.  s
) `  h )  =  ( (  _I  |`  T ) `  h
) )
24233adant2r 1178 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( ( U  o.  s ) `  h
)  =  ( (  _I  |`  T ) `  h ) )
25 simp2 957 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )
262, 5, 6, 13tendocan 31072 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( U  o.  s )  e.  E  /\  (  _I  |`  T )  e.  E  /\  ( ( U  o.  s ) `  h
)  =  ( (  _I  |`  T ) `  h ) )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  ( U  o.  s )  =  (  _I  |`  T )
)
271, 20, 22, 24, 25, 26syl131anc 1196 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  =  (  _I  |`  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   ifcif 3654    e. cmpt 4179    _I cid 4407   `'ccnv 4791    |` cres 4794    o. ccom 4796   ` cfv 5358  (class class class)co 5981   iota_crio 6439   Basecbs 13356   occoc 13424   joincjn 14288   meetcmee 14289   HLchlt 29599   LHypclh 30232   LTrncltrn 30349   trLctrl 30406   TEndoctendo 31000
This theorem is referenced by:  cdleml9  31232  erngdvlem4  31239  erngdvlem4-rN  31247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-fal 1325  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-llines 29746  df-lplanes 29747  df-lvols 29748  df-lines 29749  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-lhyp 30236  df-laut 30237  df-ldil 30352  df-ltrn 30353  df-trl 30407  df-tendo 31003
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