Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme42mgN Unicode version

Theorem cdleme42mgN 31016
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT . f preserves join: f(r  \/ s) = f(r)  \/ s, p. 115 10th line from bottom. TODO: Use instead of cdleme42mN 31015? Combine with cdleme42mN 31015? (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme41.b  |-  B  =  ( Base `  K
)
cdleme41.l  |-  .<_  =  ( le `  K )
cdleme41.j  |-  .\/  =  ( join `  K )
cdleme41.m  |-  ./\  =  ( meet `  K )
cdleme41.a  |-  A  =  ( Atoms `  K )
cdleme41.h  |-  H  =  ( LHyp `  K
)
cdleme41.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme41.d  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme41.e  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme41.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme41.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
cdleme41.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme41.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme41.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdleme42mgN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  ( F `  ( R  .\/  S ) )  =  ( ( F `  R )  .\/  ( F `  S )
) )
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, s    S, s    U, s    W, s    y, t, A, s    B, s, t, y    y, D    y, G    E, s,
y    H, s, t, y   
t,  .\/ , y    K, s, t, y    t,  .<_ , y   
t,  ./\ , y    t, P, y    t, Q, y    t, R, y    t, S, y   
t, U, y    t, W, y    x, z, A   
x, B, z    z, E, s    z, H    x,  .\/ , z    z, K    x,  .<_ , z    x,  ./\ , z    x, N, z    x, P, z   
x, Q, z    x, R, z    x, S, z   
x, U, z    x, W, z, s, t, y
Allowed substitution hints:    D( x, z, t, s)    E( x, t)    F( x, y, z, t, s)    G( x, z, t, s)    H( x)    I( x, y, z, t, s)    K( x)    N( y, t, s)    O( x, y, z, t, s)

Proof of Theorem cdleme42mgN
StepHypRef Expression
1 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  K  e.  HL )
2 hllat 29892 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  K  e.  Lat )
4 simprll 739 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  R  e.  A )
5 cdleme41.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cdleme41.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 29818 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
84, 7syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  R  e.  B )
9 simprrl 741 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  S  e.  A )
105, 6atbase 29818 . . . . 5  |-  ( S  e.  A  ->  S  e.  B )
119, 10syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  S  e.  B )
123, 8, 113jca 1134 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  ( K  e.  Lat  /\  R  e.  B  /\  S  e.  B ) )
13 cdleme41.j . . . . . 6  |-  .\/  =  ( join `  K )
145, 13latjcl 14462 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  B  /\  S  e.  B )  ->  ( R  .\/  S
)  e.  B )
15 cdleme41.f . . . . . 6  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
1615cdleme31id 30922 . . . . 5  |-  ( ( ( R  .\/  S
)  e.  B  /\  P  =  Q )  ->  ( F `  ( R  .\/  S ) )  =  ( R  .\/  S ) )
1714, 16sylan 458 . . . 4  |-  ( ( ( K  e.  Lat  /\  R  e.  B  /\  S  e.  B )  /\  P  =  Q
)  ->  ( F `  ( R  .\/  S
) )  =  ( R  .\/  S ) )
1815cdleme31id 30922 . . . . . 6  |-  ( ( R  e.  B  /\  P  =  Q )  ->  ( F `  R
)  =  R )
19183ad2antl2 1120 . . . . 5  |-  ( ( ( K  e.  Lat  /\  R  e.  B  /\  S  e.  B )  /\  P  =  Q
)  ->  ( F `  R )  =  R )
2015cdleme31id 30922 . . . . . 6  |-  ( ( S  e.  B  /\  P  =  Q )  ->  ( F `  S
)  =  S )
21203ad2antl3 1121 . . . . 5  |-  ( ( ( K  e.  Lat  /\  R  e.  B  /\  S  e.  B )  /\  P  =  Q
)  ->  ( F `  S )  =  S )
2219, 21oveq12d 6085 . . . 4  |-  ( ( ( K  e.  Lat  /\  R  e.  B  /\  S  e.  B )  /\  P  =  Q
)  ->  ( ( F `  R )  .\/  ( F `  S
) )  =  ( R  .\/  S ) )
2317, 22eqtr4d 2465 . . 3  |-  ( ( ( K  e.  Lat  /\  R  e.  B  /\  S  e.  B )  /\  P  =  Q
)  ->  ( F `  ( R  .\/  S
) )  =  ( ( F `  R
)  .\/  ( F `  S ) ) )
2412, 23sylan 458 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  /\  P  =  Q )  ->  ( F `  ( R  .\/  S ) )  =  ( ( F `  R )  .\/  ( F `  S )
) )
25 simpll 731 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  /\  P  =/=  Q )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
26 simpr 448 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  /\  P  =/=  Q )  ->  P  =/=  Q )
27 simplrl 737 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  /\  P  =/=  Q )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
28 simplrr 738 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  /\  P  =/=  Q )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
29 cdleme41.l . . . 4  |-  .<_  =  ( le `  K )
30 cdleme41.m . . . 4  |-  ./\  =  ( meet `  K )
31 cdleme41.h . . . 4  |-  H  =  ( LHyp `  K
)
32 cdleme41.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
33 cdleme41.d . . . 4  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
34 cdleme41.e . . . 4  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
35 cdleme41.g . . . 4  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
36 cdleme41.i . . . 4  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
37 cdleme41.n . . . 4  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
38 cdleme41.o . . . 4  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
395, 29, 13, 30, 6, 31, 32, 33, 34, 35, 36, 37, 38, 15cdleme42mN 31015 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  ( F `  ( R  .\/  S ) )  =  ( ( F `  R ) 
.\/  ( F `  S ) ) )
4025, 26, 27, 28, 39syl13anc 1186 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  /\  P  =/=  Q )  ->  ( F `  ( R  .\/  S ) )  =  ( ( F `  R )  .\/  ( F `  S )
) )
4124, 40pm2.61dane 2671 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  ( F `  ( R  .\/  S ) )  =  ( ( F `  R )  .\/  ( F `  S )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   A.wral 2692   ifcif 3726   class class class wbr 4199    e. cmpt 4253   ` cfv 5440  (class class class)co 6067   iota_crio 6528   Basecbs 13452   lecple 13519   joincjn 14384   meetcmee 14385   Latclat 14457   Atomscatm 29792   HLchlt 29879   LHypclh 30512
This theorem is referenced by:  cdlemg2jlemOLDN  31121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028  df-lines 30029  df-psubsp 30031  df-pmap 30032  df-padd 30324  df-lhyp 30516
  Copyright terms: Public domain W3C validator