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Theorem cdleme32fva1 30966
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Mar-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
cdleme32fva1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( F `  R )  =  [_ R  /  s ]_ N
)
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    R, s, t, y    y, H    y, K    x, R, z    z, H    z, K
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)

Proof of Theorem cdleme32fva1
StepHypRef Expression
1 simp2l 983 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  R  e.  A )
2 cdleme32.b . . . . 5  |-  B  =  ( Base `  K
)
3 cdleme32.a . . . . 5  |-  A  =  ( Atoms `  K )
42, 3atbase 29818 . . . 4  |-  ( R  e.  A  ->  R  e.  B )
51, 4syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  R  e.  B )
6 simp3 959 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  P  =/=  Q )
7 simp2r 984 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  -.  R  .<_  W )
8 cdleme32.o . . . 4  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
9 cdleme32.f . . . 4  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
108, 9cdleme31fv1s 30920 . . 3  |-  ( ( R  e.  B  /\  ( P  =/=  Q  /\  -.  R  .<_  W ) )  ->  ( F `  R )  =  [_ R  /  x ]_ O
)
115, 6, 7, 10syl12anc 1182 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( F `  R )  =  [_ R  /  x ]_ O
)
12 cdleme32.l . . 3  |-  .<_  =  ( le `  K )
13 cdleme32.j . . 3  |-  .\/  =  ( join `  K )
14 cdleme32.m . . 3  |-  ./\  =  ( meet `  K )
15 cdleme32.h . . 3  |-  H  =  ( LHyp `  K
)
16 cdleme32.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
17 cdleme32.c . . 3  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
18 cdleme32.d . . 3  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
19 cdleme32.e . . 3  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
20 cdleme32.i . . 3  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
21 cdleme32.n . . 3  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
222, 12, 13, 14, 3, 15, 16, 17, 18, 19, 20, 21, 8, 9cdleme32fva 30965 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N
)
2311, 22eqtrd 2462 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( F `  R )  =  [_ R  /  s ]_ N
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   A.wral 2692   [_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:  cdleme32fvaw  30967  cdleme41fva11  31005  cdleme42f  31008  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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