Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme42b Unicode version

Theorem cdleme42b 30667
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 6-Mar-2013.)
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
cdleme42b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, 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, 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, U, z    x, W, z, s, t, y    X, s, t, x, z
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)    X( y)

Proof of Theorem cdleme42b
StepHypRef Expression
1 cdleme41.b . . 3  |-  B  =  ( Base `  K
)
2 fvex 5539 . . 3  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2353 . 2  |-  B  e. 
_V
4 nfv 1605 . . 3  |-  F/ s ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )
5 nfcsb1v 3113 . . . . 5  |-  F/_ s [_ R  /  s ]_ N
6 nfcv 2419 . . . . 5  |-  F/_ s  .\/
7 nfcv 2419 . . . . 5  |-  F/_ s
( X  ./\  W
)
85, 6, 7nfov 5881 . . . 4  |-  F/_ s
( [_ R  /  s ]_ N  .\/  ( X 
./\  W ) )
98a1i 10 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  F/_ s (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
10 nfvd 1606 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  F/ s
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )
11 cdleme41.o . . . . 5  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
12 cdleme41.f . . . . 5  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
13 eqid 2283 . . . . 5  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )
1411, 12, 13cdleme31fv1 30580 . . . 4  |-  ( ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  (
iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
15143ad2ant2 977 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  (
iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
16 breq1 4026 . . . . . 6  |-  ( s  =  R  ->  (
s  .<_  W  <->  R  .<_  W ) )
1716notbid 285 . . . . 5  |-  ( s  =  R  ->  ( -.  s  .<_  W  <->  -.  R  .<_  W ) )
18 oveq1 5865 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  ( X  ./\ 
W ) )  =  ( R  .\/  ( X  ./\  W ) ) )
1918eqeq1d 2291 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  ( X  ./\  W ) )  =  X  <->  ( R  .\/  ( X  ./\  W
) )  =  X ) )
2017, 19anbi12d 691 . . . 4  |-  ( s  =  R  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) ) )
2120adantl 452 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  /\  s  =  R )  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) ) )
22 csbeq1a 3089 . . . . . 6  |-  ( s  =  R  ->  N  =  [_ R  /  s ]_ N )
2322oveq1d 5873 . . . . 5  |-  ( s  =  R  ->  ( N  .\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
2423a1i 10 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( s  =  R  ->  ( N 
.\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) ) )
2524imp 418 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  /\  s  =  R )  ->  ( N  .\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
26 simp1 955 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
27 simp2l 981 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
28 cdleme41.l . . . . 5  |-  .<_  =  ( le `  K )
29 cdleme41.j . . . . 5  |-  .\/  =  ( join `  K )
30 cdleme41.m . . . . 5  |-  ./\  =  ( meet `  K )
31 cdleme41.a . . . . 5  |-  A  =  ( Atoms `  K )
32 cdleme41.h . . . . 5  |-  H  =  ( LHyp `  K
)
33 cdleme41.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
34 cdleme41.d . . . . 5  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
35 cdleme41.e . . . . 5  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
36 cdleme41.g . . . . 5  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
37 cdleme41.i . . . . 5  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
38 cdleme41.n . . . . 5  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
391, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 11, 12cdleme32fvcl 30629 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( F `  X
)  e.  B )
4026, 27, 39syl2anc 642 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  e.  B
)
41 simp3ll 1026 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  R  e.  A )
42 simp3lr 1027 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  R  .<_  W )
43 simp3r 984 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( R  .\/  ( X  ./\  W
) )  =  X )
4442, 43jca 518 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )
454, 9, 10, 15, 21, 25, 40, 41, 44riotasv2d 6349 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  /\  B  e. 
_V )  ->  ( F `  X )  =  ( [_ R  /  s ]_ N  .\/  ( X  ./\  W
) ) )
463, 45mpan2 652 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   F/_wnfc 2406    =/= wne 2446   A.wral 2543   _Vcvv 2788   [_csb 3081   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme42e  30668  cdleme48fv  30688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177
  Copyright terms: Public domain W3C validator