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Theorem cdleml9 31795
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
cdleml9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
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 cdleml9
StepHypRef Expression
1 cdleml6.b . . . 4  |-  B  =  ( Base `  K
)
2 cdleml6.h . . . 4  |-  H  =  ( LHyp `  K
)
3 cdleml6.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
4 cdleml6.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
5 cdleml6.o . . . 4  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
61, 2, 3, 4, 5tendo1ne0 31639 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  .0.  )
763ad2ant1 976 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
(  _I  |`  T )  =/=  .0.  )
8 cdleml6.j . . . . . . 7  |-  .\/  =  ( join `  K )
9 cdleml6.m . . . . . . 7  |-  ./\  =  ( meet `  K )
10 cdleml6.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
11 cdleml6.p . . . . . . 7  |-  Q  =  ( ( oc `  K ) `  W
)
12 cdleml6.z . . . . . . 7  |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( (
h `  Q )  .\/  ( R `  (
b  o.  `' ( s `  h ) ) ) ) )
13 cdleml6.y . . . . . . 7  |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
14 cdleml6.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  -> 
( z `  Q
)  =  Y ) )
15 cdleml6.u . . . . . . 7  |-  U  =  ( g  e.  T  |->  if ( ( s `
 h )  =  h ,  g ,  X ) )
161, 8, 9, 2, 3, 10, 11, 12, 13, 14, 15, 4, 5cdleml8 31794 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  =  (  _I  |`  T ) )
1716adantr 451 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  /\  U  =  .0.  )  ->  ( U  o.  s
)  =  (  _I  |`  T ) )
18 coeq1 4857 . . . . . 6  |-  ( U  =  .0.  ->  ( U  o.  s )  =  (  .0.  o.  s
) )
19 simp1 955 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
20 simp3l 983 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
s  e.  E )
211, 2, 3, 4, 5tendo0mul 31637 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E
)  ->  (  .0.  o.  s )  =  .0.  )
2219, 20, 21syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
(  .0.  o.  s
)  =  .0.  )
2318, 22sylan9eqr 2350 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  /\  U  =  .0.  )  ->  ( U  o.  s
)  =  .0.  )
2417, 23eqtr3d 2330 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  /\  U  =  .0.  )  ->  (  _I  |`  T )  =  .0.  )
2524ex 423 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  =  .0. 
->  (  _I  |`  T )  =  .0.  ) )
2625necon3d 2497 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( (  _I  |`  T )  =/=  .0.  ->  U  =/=  .0.  ) )
277, 26mpd 14 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   ifcif 3578    e. cmpt 4093    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   occoc 13232   joincjn 14094   meetcmee 14095   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  erngdvlem4  31802  erngdvlem4-rN  31810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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