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Theorem cdleml9 30340
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 30184 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  .0.  )
763ad2ant1 981 . 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 30339 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  =  (  _I  |`  T ) )
1716adantr 453 . . . . 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 4829 . . . . . 6  |-  ( U  =  .0.  ->  ( U  o.  s )  =  (  .0.  o.  s
) )
19 simp1 960 . . . . . . 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 988 . . . . . . 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 30182 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E
)  ->  (  .0.  o.  s )  =  .0.  )
2219, 20, 21syl2anc 645 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
(  .0.  o.  s
)  =  .0.  )
2318, 22sylan9eqr 2312 . . . . 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 2292 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  /\  U  =  .0.  )  ->  (  _I  |`  T )  =  .0.  )
2524ex 425 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  =  .0. 
->  (  _I  |`  T )  =  .0.  ) )
2625necon3d 2459 . 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 16 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   ifcif 3539    e. cmpt 4051    _I cid 4276   `'ccnv 4660    |` cres 4663    o. ccom 4665   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13110   occoc 13178   joincjn 14040   meetcmee 14041   HLchlt 28707   LHypclh 29340   LTrncltrn 29457   trLctrl 29514   TEndoctendo 30108
This theorem is referenced by:  erngdvlem4  30347  erngdvlem4-rN  30355
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tendo 30111
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