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Theorem cdleml9 34967
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 34811 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  .0.  )
763ad2ant1 1009 . 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 34966 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  =  (  _I  |`  T ) )
1716adantr 465 . . . . 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 5106 . . . . . 6  |-  ( U  =  .0.  ->  ( U  o.  s )  =  (  .0.  o.  s
) )
19 simp1 988 . . . . . . 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 1016 . . . . . . 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 34809 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E
)  ->  (  .0.  o.  s )  =  .0.  )
2219, 20, 21syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
(  .0.  o.  s
)  =  .0.  )
2318, 22sylan9eqr 2517 . . . . 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 2497 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  /\  U  =  .0.  )  ->  (  _I  |`  T )  =  .0.  )
2524ex 434 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  =  .0. 
->  (  _I  |`  T )  =  .0.  ) )
2625necon3d 2676 . 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 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   ifcif 3900    |-> cmpt 4459    _I cid 4740   `'ccnv 4948    |` cres 4951    o. ccom 4953   ` cfv 5527   iota_crio 6161  (class class class)co 6201   Basecbs 14293   occoc 14366   joincjn 15234   meetcmee 15235   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141   TEndoctendo 34735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-undef 6903  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142  df-tendo 34738
This theorem is referenced by:  erngdvlem4  34974  erngdvlem4-rN  34982
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