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Theorem cdleme42b 31006
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 5728 . . 3  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2500 . 2  |-  B  e. 
_V
4 nfv 1629 . . 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 3270 . . . . 5  |-  F/_ s [_ R  /  s ]_ N
6 nfcv 2566 . . . . 5  |-  F/_ s  .\/
7 nfcv 2566 . . . . 5  |-  F/_ s
( X  ./\  W
)
85, 6, 7nfov 6090 . . . 4  |-  F/_ s
( [_ R  /  s ]_ N  .\/  ( X 
./\  W ) )
98a1i 11 . . 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 1630 . . 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 2430 . . . . 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 30919 . . . 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 979 . . 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 4202 . . . . . 6  |-  ( s  =  R  ->  (
s  .<_  W  <->  R  .<_  W ) )
1716notbid 286 . . . . 5  |-  ( s  =  R  ->  ( -.  s  .<_  W  <->  -.  R  .<_  W ) )
18 oveq1 6074 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  ( X  ./\ 
W ) )  =  ( R  .\/  ( X  ./\  W ) ) )
1918eqeq1d 2438 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  ( X  ./\  W ) )  =  X  <->  ( R  .\/  ( X  ./\  W
) )  =  X ) )
2017, 19anbi12d 692 . . . 4  |-  ( s  =  R  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  R  .<_  W  /\  ( R  .\/  ( X  ./\  W ) )  =  X ) ) )
2120adantl 453 . . 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 3246 . . . . 5  |-  ( s  =  R  ->  N  =  [_ R  /  s ]_ N )
2322oveq1d 6082 . . . 4  |-  ( s  =  R  ->  ( N  .\/  ( X  ./\  W ) )  =  (
[_ R  /  s ]_ N  .\/  ( X 
./\  W ) ) )
2423adantl 453 . . 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 ) ) )
25 simp1 957 . . . 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 ) ) )
26 simp2l 983 . . . 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 )
27 cdleme41.l . . . . 5  |-  .<_  =  ( le `  K )
28 cdleme41.j . . . . 5  |-  .\/  =  ( join `  K )
29 cdleme41.m . . . . 5  |-  ./\  =  ( meet `  K )
30 cdleme41.a . . . . 5  |-  A  =  ( Atoms `  K )
31 cdleme41.h . . . . 5  |-  H  =  ( LHyp `  K
)
32 cdleme41.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
33 cdleme41.d . . . . 5  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
34 cdleme41.e . . . . 5  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
35 cdleme41.g . . . . 5  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
36 cdleme41.i . . . . 5  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
37 cdleme41.n . . . . 5  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
381, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 11, 12cdleme32fvcl 30968 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( F `  X
)  e.  B )
3925, 26, 38syl2anc 643 . . 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
)
40 simp3ll 1028 . . 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 )
41 simp3lr 1029 . . . 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 )
42 simp3r 986 . . . 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 )
4341, 42jca 519 . . 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 ) )
444, 9, 10, 15, 21, 24, 39, 40, 43riotasv2d 6580 . 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
) ) )
453, 44mpan2 653 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   F/_wnfc 2553    =/= wne 2593   A.wral 2692   _Vcvv 2943   [_csb 3238   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   Atomscatm 29792   HLchlt 29879   LHypclh 30512
This theorem is referenced by:  cdleme42e  31007  cdleme48fv  31027
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
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