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Theorem cdlemefr32fvaN 30523
Description: Part of proof of Lemma E in [Crawley] p. 113. Value of  F at an atom not under  W. TODO: FIX COMMENT (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefr27.b  |-  B  =  ( Base `  K
)
cdlemefr27.l  |-  .<_  =  ( le `  K )
cdlemefr27.j  |-  .\/  =  ( join `  K )
cdlemefr27.m  |-  ./\  =  ( meet `  K )
cdlemefr27.a  |-  A  =  ( Atoms `  K )
cdlemefr27.h  |-  H  =  ( LHyp `  K
)
cdlemefr27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemefr27.c  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdlemefr27.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
cdleme29fr.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
Assertion
Ref Expression
cdlemefr32fvaN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  [_ R  /  x ]_ O  = 
[_ R  /  s ]_ N )
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, s    U, s    W, s, z    H, s    K, s    x, z, A, s    B, s, x, z    z, H   
x,  .\/ , z    z, K   
x,  .<_ , z    x,  ./\ , z    x, N, z    z, P   
z, Q    x, R, z    x, W, z
Allowed substitution hints:    C( x, z, s)    P( x)    Q( x)    U( x, z)    H( x)    I( x, z, s)    K( x)    N( s)    O( x, z, s)

Proof of Theorem cdlemefr32fvaN
StepHypRef Expression
1 cdlemefr27.b . 2  |-  B  =  ( Base `  K
)
2 cdlemefr27.l . 2  |-  .<_  =  ( le `  K )
3 cdlemefr27.j . 2  |-  .\/  =  ( join `  K )
4 cdlemefr27.m . 2  |-  ./\  =  ( meet `  K )
5 cdlemefr27.a . 2  |-  A  =  ( Atoms `  K )
6 cdlemefr27.h . 2  |-  H  =  ( LHyp `  K
)
7 breq1 4156 . . 3  |-  ( s  =  R  ->  (
s  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  Q ) ) )
87notbid 286 . 2  |-  ( s  =  R  ->  ( -.  s  .<_  ( P 
.\/  Q )  <->  -.  R  .<_  ( P  .\/  Q
) ) )
9 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
10 simp12l 1070 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) ) )  ->  P  e.  A )
11 simp13l 1072 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) ) )  ->  Q  e.  A )
12 simp3l 985 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) ) )  -> 
s  e.  A )
13 simp3rr 1031 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) ) )  ->  -.  s  .<_  ( P 
.\/  Q ) )
14 simp2 958 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) ) )  ->  P  =/=  Q )
15 cdlemefr27.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
16 cdlemefr27.c . . . 4  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
17 cdlemefr27.n . . . 4  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
181, 2, 3, 4, 5, 6, 15, 16, 17cdlemefr27cl 30517 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  N  e.  B
)
199, 10, 11, 12, 13, 14, 18syl33anc 1199 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) ) )  ->  N  e.  B )
201, 2, 3, 4, 5, 6, 15, 16, 17cdlemefr32snb 30519 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  [_ R  /  s ]_ N  e.  B )
21 cdleme29fr.o . 2  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
221, 2, 3, 4, 5, 6, 8, 19, 20, 21cdlemefrs32fva 30514 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  [_ R  /  x ]_ O  = 
[_ R  /  s ]_ N )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   [_csb 3194   ifcif 3682   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   iota_crio 6478   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   Atomscatm 29378   HLchlt 29465   LHypclh 30098
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102
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