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Theorem cdleme32d 30930
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
cdleme32.b  |-  B  =  ( Base `  K
)
cdleme32.l  |-  .<_  =  ( le `  K )
cdleme32.j  |-  .\/  =  ( join `  K )
cdleme32.m  |-  ./\  =  ( meet `  K )
cdleme32.a  |-  A  =  ( Atoms `  K )
cdleme32.h  |-  H  =  ( LHyp `  K
)
cdleme32.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme32.c  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme32.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme32.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme32.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
cdleme32.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
cdleme32.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme32.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdleme32d  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  ( F `  X )  .<_  ( F `
 Y ) )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z   
y, C    D, s,
y, z    y, E    H, s, t    .\/ , s,
t, x, y, z    K, s, t    .<_ , s, t, x, y, z    ./\ , s,
t, x, y, z   
x, N, z    P, s, t, x, y, z    Q, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    X, s, t, x, z   
y, H    y, K    y, Y    z, H    z, K    Y, s, t, x, z
Allowed substitution hints:    C( x, z, t, s)    D( x, t)    E( x, z, t, s)    F( x, y, 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 cdleme32d
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp21 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  X  e.  B )
3 simp23r 1079 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  -.  X  .<_  W )
4 cdleme32.b . . . 4  |-  B  =  ( Base `  K
)
5 cdleme32.l . . . 4  |-  .<_  =  ( le `  K )
6 cdleme32.j . . . 4  |-  .\/  =  ( join `  K )
7 cdleme32.m . . . 4  |-  ./\  =  ( meet `  K )
8 cdleme32.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdleme32.h . . . 4  |-  H  =  ( LHyp `  K
)
104, 5, 6, 7, 8, 9lhpmcvr2 30510 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) )
111, 2, 3, 10syl12anc 1182 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  (
s  .\/  ( X  ./\ 
W ) )  =  X ) )
12 nfv 1626 . . 3  |-  F/ s ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )
13 cdleme32.f . . . . . 6  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
14 nfcv 2544 . . . . . . 7  |-  F/_ s B
15 nfv 1626 . . . . . . . 8  |-  F/ s ( P  =/=  Q  /\  -.  x  .<_  W )
16 cdleme32.o . . . . . . . . 9  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
17 nfra1 2720 . . . . . . . . . 10  |-  F/ s A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )
1817, 14nfriota 6522 . . . . . . . . 9  |-  F/_ s
( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
1916, 18nfcxfr 2541 . . . . . . . 8  |-  F/_ s O
20 nfcv 2544 . . . . . . . 8  |-  F/_ s
x
2115, 19, 20nfif 3727 . . . . . . 7  |-  F/_ s if ( ( P  =/= 
Q  /\  -.  x  .<_  W ) ,  O ,  x )
2214, 21nfmpt 4261 . . . . . 6  |-  F/_ s
( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
2313, 22nfcxfr 2541 . . . . 5  |-  F/_ s F
24 nfcv 2544 . . . . 5  |-  F/_ s X
2523, 24nffv 5698 . . . 4  |-  F/_ s
( F `  X
)
26 nfcv 2544 . . . 4  |-  F/_ s  .<_
27 nfcv 2544 . . . . 5  |-  F/_ s Y
2823, 27nffv 5698 . . . 4  |-  F/_ s
( F `  Y
)
2925, 26, 28nfbr 4220 . . 3  |-  F/ s ( F `  X
)  .<_  ( F `  Y )
30 simpl1 960 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
31 simpl2 961 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )
32 simprl 733 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  s  e.  A )
33 simprrl 741 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  -.  s  .<_  W )
3432, 33jca 519 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  ( s  e.  A  /\  -.  s  .<_  W ) )
35 simprrr 742 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  ( s  .\/  ( X  ./\  W
) )  =  X )
36 simpl3 962 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  X  .<_  Y )
37 cdleme32.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
38 cdleme32.c . . . . . 6  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
39 cdleme32.d . . . . . 6  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
40 cdleme32.e . . . . . 6  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
41 cdleme32.i . . . . . 6  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
42 cdleme32.n . . . . . 6  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
434, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 16, 13cdleme32c 30929 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  ( s  .\/  ( X  ./\  W ) )  =  X  /\  X  .<_  Y ) )  ->  ( F `  X )  .<_  ( F `
 Y ) )
4430, 31, 34, 35, 36, 43syl113anc 1196 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  ( F `  X )  .<_  ( F `
 Y ) )
4544exp32 589 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  ( s  e.  A  ->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
( F `  X
)  .<_  ( F `  Y ) ) ) )
4612, 29, 45rexlimd 2791 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  ( E. s  e.  A  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
( F `  X
)  .<_  ( F `  Y ) ) )
4711, 46mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  ( F `  X )  .<_  ( F `
 Y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   E.wrex 2671   ifcif 3703   class class class wbr 4176    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   iota_crio 6505   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   HLchlt 29837   LHypclh 30470
This theorem is referenced by:  cdleme32le  30933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474
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