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Theorem cdlemk55u1 34457
Description: Lemma for cdlemk55u 34458. (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
cdlemk55u1  |-  ( ( ( ( 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 ) ) )
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 cdlemk55u1
StepHypRef Expression
1 simp11 1036 . . 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 ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp21l 1123 . . 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 ) )  -> 
( R `  F
)  =  ( R `
 N ) )
3 simp12 1037 . . . 4  |-  ( ( ( ( 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 ) )  ->  F  e.  T )
4 simp13 1038 . . . . 5  |-  ( ( ( ( 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 ) )  ->  N  e.  T )
5 simp21r 1124 . . . . 5  |-  ( ( ( ( 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 ) )  ->  F  =/=  N )
6 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
7 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
8 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
106, 7, 8, 9trlnid 33670 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( F  =/=  N  /\  ( R `  F )  =  ( R `  N ) ) )  ->  F  =/=  (  _I  |`  B ) )
111, 3, 4, 5, 2, 10syl122anc 1274 . . . 4  |-  ( ( ( ( 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 ) )  ->  F  =/=  (  _I  |`  B ) )
123, 11, 43jca 1186 . . 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 ) )  -> 
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )
)
13 simp22 1040 . . 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 ) )  ->  G  e.  T )
14 simp23 1041 . . 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 ) )  ->  I  e.  T )
15 simp3 1008 . . 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 ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
16 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
17 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
18 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
19 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
20 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
21 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
22 cdlemk5.x . . . 4  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
236, 16, 17, 18, 19, 7, 8, 9, 20, 21, 22cdlemk55 34453 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  [_ ( G  o.  I
)  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
241, 2, 12, 13, 14, 15, 23syl231anc 1285 . 2  |-  ( ( ( ( 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 ) )  ->  [_ ( G  o.  I
)  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
257, 8ltrnco 34211 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  I  e.  T
)  ->  ( G  o.  I )  e.  T
)
261, 13, 14, 25syl3anc 1265 . . 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 ) )  -> 
( G  o.  I
)  e.  T )
27 cdlemk5.u . . . 4  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
2822, 27cdlemk40f 34411 . . 3  |-  ( ( F  =/=  N  /\  ( G  o.  I
)  e.  T )  ->  ( U `  ( G  o.  I
) )  =  [_ ( G  o.  I
)  /  g ]_ X )
295, 26, 28syl2anc 666 . 2  |-  ( ( ( ( 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 )
)  =  [_ ( G  o.  I )  /  g ]_ X
)
3022, 27cdlemk40f 34411 . . . 4  |-  ( ( F  =/=  N  /\  G  e.  T )  ->  ( U `  G
)  =  [_ G  /  g ]_ X
)
315, 13, 30syl2anc 666 . . 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
)  =  [_ G  /  g ]_ X
)
3222, 27cdlemk40f 34411 . . . 4  |-  ( ( F  =/=  N  /\  I  e.  T )  ->  ( U `  I
)  =  [_ I  /  g ]_ X
)
335, 14, 32syl2anc 666 . . 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 `  I
)  =  [_ I  /  g ]_ X
)
3431, 33coeq12d 5016 . 2  |-  ( ( ( ( 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.  ( U `  I )
)  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
3524, 29, 343eqtr4d 2474 1  |-  ( ( ( ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   [_csb 3396   ifcif 3910   class class class wbr 4421    |-> cmpt 4480    _I cid 4761   `'ccnv 4850    |` cres 4853    o. ccom 4855   ` cfv 5599   iota_crio 6264  (class class class)co 6303   Basecbs 15114   lecple 15190   joincjn 16182   meetcmee 16183   Atomscatm 32754   HLchlt 32841   LHypclh 33474   LTrncltrn 33591   trLctrl 33649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-riotaBAD 32450
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-undef 7026  df-map 7480  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32667  df-ol 32669  df-oml 32670  df-covers 32757  df-ats 32758  df-atl 32789  df-cvlat 32813  df-hlat 32842  df-llines 32988  df-lplanes 32989  df-lvols 32990  df-lines 32991  df-psubsp 32993  df-pmap 32994  df-padd 33286  df-lhyp 33478  df-laut 33479  df-ldil 33594  df-ltrn 33595  df-trl 33650
This theorem is referenced by:  cdlemk55u  34458
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