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Theorem cdlemk55u 30322
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120.  G,  I stand for g, h.  X represents tau. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
cdlemk5.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
Assertion
Ref Expression
cdlemk55u  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  ( G  o.  I
) )  =  ( ( U `  G
)  o.  ( U `
 I ) ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, G, z    ./\ , b, z    .<_ , b    z,
g,  .<_    .\/ , b, z    A, b, g, z    B, b, z    F, b, g, z   
z, G    H, b,
g, z    K, b,
g, z    N, b,
g, z    P, b,
z    R, b, z    T, b, z    W, b, g, z    z, Y    G, b    I, b, g, z
Allowed substitution hints:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk55u
StepHypRef Expression
1 simpr 449 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  F  =  N )
2 simp11 990 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp22 994 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
4 simp23 995 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  I  e.  T )
5 cdlemk5.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
6 cdlemk5.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
75, 6ltrnco 30075 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  I  e.  T
)  ->  ( G  o.  I )  e.  T
)
82, 3, 4, 7syl3anc 1187 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G  o.  I )  e.  T
)
98adantr 453 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( G  o.  I )  e.  T )
10 cdlemk5.x . . . . 5  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
11 cdlemk5.u . . . . 5  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
1210, 11cdlemk40t 30274 . . . 4  |-  ( ( F  =  N  /\  ( G  o.  I
)  e.  T )  ->  ( U `  ( G  o.  I
) )  =  ( G  o.  I ) )
131, 9, 12syl2anc 645 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  ( G  o.  I ) )  =  ( G  o.  I
) )
14 simpl22 1039 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  G  e.  T )
1510, 11cdlemk40t 30274 . . . . 5  |-  ( ( F  =  N  /\  G  e.  T )  ->  ( U `  G
)  =  G )
161, 14, 15syl2anc 645 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  G )  =  G )
17 simpl23 1040 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  I  e.  T )
1810, 11cdlemk40t 30274 . . . . 5  |-  ( ( F  =  N  /\  I  e.  T )  ->  ( U `  I
)  =  I )
191, 17, 18syl2anc 645 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  I )  =  I )
2016, 19coeq12d 4836 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  (
( U `  G
)  o.  ( U `
 I ) )  =  ( G  o.  I ) )
2113, 20eqtr4d 2293 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  ( G  o.  I ) )  =  ( ( U `  G )  o.  ( U `  I )
) )
22 simpl1 963 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T
) )
23 simpl21 1038 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  ( R `  F )  =  ( R `  N ) )
24 simpr 449 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  F  =/=  N )
2523, 24jca 520 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N ) )
26 simpl22 1039 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  G  e.  T )
27 simpl23 1040 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  I  e.  T )
28 simpl3 965 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
29 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
30 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
31 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
32 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
33 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
34 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
35 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
36 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
3729, 30, 31, 32, 33, 5, 6, 34, 35, 36, 10, 11cdlemk55u1 30321 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( ( R `  F )  =  ( R `  N )  /\  F  =/=  N
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  ( G  o.  I )
)  =  ( ( U `  G )  o.  ( U `  I ) ) )
3822, 25, 26, 27, 28, 37syl131anc 1200 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  ( U `  ( G  o.  I ) )  =  ( ( U `  G )  o.  ( U `  I )
) )
3921, 38pm2.61dane 2499 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T  /\  I  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  ( G  o.  I
) )  =  ( ( U `  G
)  o.  ( U `
 I ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   ifcif 3539   class class class wbr 3997    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   lecple 13177   joincjn 14040   meetcmee 14041   Atomscatm 28620   HLchlt 28707   LHypclh 29340   LTrncltrn 29457   trLctrl 29514
This theorem is referenced by:  cdlemk56  30327
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-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
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