Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemeg49le Structured version   Unicode version

Theorem cdlemeg49le 31370
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 9-Apr-2013.)
Hypotheses
Ref Expression
cdlemef47.b  |-  B  =  ( Base `  K
)
cdlemef47.l  |-  .<_  =  ( le `  K )
cdlemef47.j  |-  .\/  =  ( join `  K )
cdlemef47.m  |-  ./\  =  ( meet `  K )
cdlemef47.a  |-  A  =  ( Atoms `  K )
cdlemef47.h  |-  H  =  ( LHyp `  K
)
cdlemef47.v  |-  V  =  ( ( Q  .\/  P )  ./\  W )
cdlemef47.n  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
cdlemefs47.o  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
cdlemef47.g  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
Assertion
Ref Expression
cdlemeg49le  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( G `  X )  .<_  ( G `
 Y ) )
Distinct variable groups:    a, b,
c, u, v, A    B, a, b, c, u, v    H, a, b, c, u, v    .\/ , a,
b, c, u, v    K, a, b, c, u, v    .<_ , a, b, c, u, v    ./\ , a,
b, c, u, v    N, a, b, c, u    O, a, b, c    P, a, b, c, u, v    Q, a, b, c, u, v    V, a, b, c, u, v    W, a, b, c, u, v    X, a, c, u, v    Y, a, b, c, u, v
Allowed substitution hints:    G( v, u, a, b, c)    N( v)    O( v, u)    X( b)

Proof of Theorem cdlemeg49le
StepHypRef Expression
1 simp11 988 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp13 990 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3 simp12 989 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp2 959 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  e.  B  /\  Y  e.  B ) )
5 simp3 960 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  .<_  Y )
6 cdlemef47.b . . 3  |-  B  =  ( Base `  K
)
7 cdlemef47.l . . 3  |-  .<_  =  ( le `  K )
8 cdlemef47.j . . 3  |-  .\/  =  ( join `  K )
9 cdlemef47.m . . 3  |-  ./\  =  ( meet `  K )
10 cdlemef47.a . . 3  |-  A  =  ( Atoms `  K )
11 cdlemef47.h . . 3  |-  H  =  ( LHyp `  K
)
12 cdlemef47.v . . 3  |-  V  =  ( ( Q  .\/  P )  ./\  W )
13 vex 2961 . . . 4  |-  u  e. 
_V
14 cdlemef47.n . . . . 5  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
15 eqid 2438 . . . . 5  |-  ( ( u  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  u )  ./\  W
) ) )  =  ( ( u  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  u )  ./\  W
) ) )
1614, 15cdleme31sc 31243 . . . 4  |-  ( u  e.  _V  ->  [_ u  /  v ]_ N  =  ( ( u 
.\/  V )  ./\  ( P  .\/  ( ( Q  .\/  u ) 
./\  W ) ) ) )
1713, 16ax-mp 8 . . 3  |-  [_ u  /  v ]_ N  =  ( ( u 
.\/  V )  ./\  ( P  .\/  ( ( Q  .\/  u ) 
./\  W ) ) )
18 cdlemefs47.o . . 3  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
19 eqid 2438 . . 3  |-  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) )  =  (
iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q  .\/  P
) )  ->  b  =  O ) )
20 eqid 2438 . . 3  |-  if ( u  .<_  ( Q  .\/  P ) ,  (
iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q  .\/  P
) )  ->  b  =  O ) ) , 
[_ u  /  v ]_ N )  =  if ( u  .<_  ( Q 
.\/  P ) ,  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q  .\/  P
) )  ->  b  =  O ) ) , 
[_ u  /  v ]_ N )
21 eqid 2438 . . 3  |-  ( iota_ c  e.  B A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u  .\/  ( a 
./\  W ) )  =  a )  -> 
c  =  ( if ( u  .<_  ( Q 
.\/  P ) ,  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q  .\/  P
) )  ->  b  =  O ) ) , 
[_ u  /  v ]_ N )  .\/  (
a  ./\  W )
) ) )  =  ( iota_ c  e.  B A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) )
22 cdlemef47.g . . 3  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
236, 7, 8, 9, 10, 11, 12, 17, 14, 18, 19, 20, 21, 22cdleme32le 31306 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( G `  X )  .<_  ( G `
 Y ) )
241, 2, 3, 4, 5, 23syl311anc 1199 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( G `  X )  .<_  ( G `
 Y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958   [_csb 3253   ifcif 3741   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  cdlemeg49lebilem  31398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847
  Copyright terms: Public domain W3C validator