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Theorem cdlemefr32fva1 30892
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 29-Mar-2013.)
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 ) ) ) )
cdleme29fr.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdlemefr32fva1  |-  ( ( ( ( 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
) )  ->  ( F `  R )  =  [_ 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    x, P    x, Q
Allowed substitution hints:    C( x, z, s)    U( x, z)    F( x, z, s)    H( x)    I( x, z, s)    K( x)    N( s)    O( x, z, s)

Proof of Theorem cdlemefr32fva1
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 4175 . . 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 30885 . . 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 30887 . 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 ) ) ) )
22 cdleme29fr.f . 2  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
231, 2, 3, 4, 5, 6, 8, 19, 20, 21, 22cdlemefrs32fva1 30883 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
) )  ->  ( F `  R )  =  [_ R  /  s ]_ N )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   [_csb 3211   ifcif 3699   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdlemefr31fv1  30893
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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