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Theorem cdlemeg49le 35708
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 1026 . 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 1028 . 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 1027 . 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 997 . 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 998 . 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 3121 . . . 4  |-  u  e. 
_V
14 cdlemef47.n . . . . 5  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
15 eqid 2467 . . . . 5  |-  ( ( u  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  u )  ./\  W
) ) )  =  ( ( u  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  u )  ./\  W
) ) )
1614, 15cdleme31sc 35581 . . . 4  |-  ( u  e.  _V  ->  [_ u  /  v ]_ N  =  ( ( u 
.\/  V )  ./\  ( P  .\/  ( ( Q  .\/  u ) 
./\  W ) ) ) )
1713, 16ax-mp 5 . . 3  |-  [_ u  /  v ]_ N  =  ( ( u 
.\/  V )  ./\  ( P  .\/  ( ( Q  .\/  u ) 
./\  W ) ) )
18 cdlemefs47.o . . 3  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
19 eqid 2467 . . 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 2467 . . 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 2467 . . 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 35644 . 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 1242 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118   [_csb 3440   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594   iota_crio 6255  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Atomscatm 34461   HLchlt 34548   LHypclh 35181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-undef 7014  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185
This theorem is referenced by:  cdlemeg49lebilem  35736
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