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Theorem cdlemk55u 37105
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 459 . . . 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 1024 . . . . . 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 1028 . . . . . 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 1029 . . . . . 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 36858 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  I  e.  T
)  ->  ( G  o.  I )  e.  T
)
82, 3, 4, 7syl3anc 1226 . . . . 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 463 . . . 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 37057 . . . 4  |-  ( ( F  =  N  /\  ( G  o.  I
)  e.  T )  ->  ( U `  ( G  o.  I
) )  =  ( G  o.  I ) )
131, 9, 12syl2anc 659 . . 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 1073 . . . . 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 37057 . . . . 5  |-  ( ( F  =  N  /\  G  e.  T )  ->  ( U `  G
)  =  G )
161, 14, 15syl2anc 659 . . . 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 1074 . . . . 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 37057 . . . . 5  |-  ( ( F  =  N  /\  I  e.  T )  ->  ( U `  I
)  =  I )
191, 17, 18syl2anc 659 . . . 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 5080 . . 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 2426 . 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 997 . . 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 1072 . . . 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 459 . . . 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 530 . . 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 1073 . . 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 1074 . . 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 999 . . 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 37104 . . 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 1239 . 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 2700 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   ifcif 3857   class class class wbr 4367    |-> cmpt 4425    _I cid 4704   `'ccnv 4912    |` cres 4915    o. ccom 4917   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   meetcmee 15691   Atomscatm 35401   HLchlt 35488   LHypclh 36121   LTrncltrn 36238   trLctrl 36296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-riotaBAD 35097
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-undef 6920  df-map 7340  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-p1 15787  df-lat 15793  df-clat 15855  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489  df-llines 35635  df-lplanes 35636  df-lvols 35637  df-lines 35638  df-psubsp 35640  df-pmap 35641  df-padd 35933  df-lhyp 36125  df-laut 36126  df-ldil 36241  df-ltrn 36242  df-trl 36297
This theorem is referenced by:  cdlemk56  37110
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