Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk39u Unicode version

Theorem cdlemk39u 30424
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by  ( U `  G ). (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
cdlemk39u  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  G
) )  .<_  ( R `
 G ) )
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
Allowed substitution hints:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk39u
StepHypRef Expression
1 simpr 449 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  F  =  N )
2 simpl2r 1011 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  G  e.  T )
3 cdlemk5.x . . . . . 6  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
4 cdlemk5.u . . . . . 6  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
53, 4cdlemk40t 30374 . . . . 5  |-  ( ( F  =  N  /\  G  e.  T )  ->  ( U `  G
)  =  G )
61, 2, 5syl2anc 644 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  G )  =  G )
76fveq2d 5489 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( R `  ( U `  G ) )  =  ( R `  G
) )
8 simp11l 1068 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
9 hllat 28820 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
108, 9syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
11 simp11 987 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
12 simp2r 984 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
13 cdlemk5.b . . . . . . 7  |-  B  =  ( Base `  K
)
14 cdlemk5.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
15 cdlemk5.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
16 cdlemk5.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
1713, 14, 15, 16trlcl 29620 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  B
)
1811, 12, 17syl2anc 644 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  e.  B
)
19 cdlemk5.l . . . . . 6  |-  .<_  =  ( le `  K )
2013, 19latref 14153 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R `  G )  e.  B )  -> 
( R `  G
)  .<_  ( R `  G ) )
2110, 18, 20syl2anc 644 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  .<_  ( R `
 G ) )
2221adantr 453 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( R `  G )  .<_  ( R `  G
) )
237, 22eqbrtrd 4044 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( R `  ( U `  G ) )  .<_  ( R `  G ) )
24 simpl1 960 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T
) )
25 simpl2l 1010 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  ( R `  F )  =  ( R `  N ) )
26 simpr 449 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  F  =/=  N )
27 simpl2r 1011 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  G  e.  T )
28 simpl3 962 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
29 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
30 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
31 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
32 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
33 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
3413, 19, 29, 30, 31, 14, 15, 16, 32, 33, 3, 4cdlemk39u1 30423 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  N  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  G
) )  .<_  ( R `
 G ) )
3524, 25, 26, 27, 28, 34syl131anc 1197 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/= 
N )  ->  ( R `  ( U `  G ) )  .<_  ( R `  G ) )
3623, 35pm2.61dane 2525 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  G
) )  .<_  ( R `
 G ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   ifcif 3566   class class class wbr 4024    e. cmpt 4078    _I cid 4303   `'ccnv 4687    |` cres 4690    o. ccom 4692   ` cfv 5221  (class class class)co 5819   iota_crio 6290   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Latclat 14145   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem is referenced by:  cdlemk56  30427
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
  Copyright terms: Public domain W3C validator