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Theorem cdlemefs32fvaN 31156
Description: Part of proof of Lemma E in [Crawley] p. 113. Value of  F at an atom not under  W. TODO: FIX COMMENT TODO: consolidate uses of lhpmat 30764 here and elsewhere, and presence/absence of  s 
.<_  ( P  .\/  Q
) term. Also, why can proof be shortened with cdleme27cl 31100? What is difference from cdlemefs27cl 31147? (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefs32.b  |-  B  =  ( Base `  K
)
cdlemefs32.l  |-  .<_  =  ( le `  K )
cdlemefs32.j  |-  .\/  =  ( join `  K )
cdlemefs32.m  |-  ./\  =  ( meet `  K )
cdlemefs32.a  |-  A  =  ( Atoms `  K )
cdlemefs32.h  |-  H  =  ( LHyp `  K
)
cdlemefs32.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemefs32.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs32.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemefs32.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
cdlemefs32.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
cdleme29fs.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
Assertion
Ref Expression
cdlemefs32fvaN  |-  ( ( ( ( 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:    t, s, x, y, z, A    B, s, t, x, y, z   
y, D    y, E    H, s, t, y    .\/ , s, t, x, y, z    K, s, t, y    .<_ , s, t, x, y, z    ./\ , s, t, x, y, z    x, N, z    P, s, t, y, z    Q, s, t, y, z    R, s, t, y    t, U, y    W, s, t, x, y, z    D, s    z, H    z, K    z, R, x
Allowed substitution hints:    C( x, y, z, t, s)    D( x, z, t)    P( x)    Q( x)    U( x, z, s)    E( x, 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 cdlemefs32fvaN
StepHypRef Expression
1 cdlemefs32.b . 2  |-  B  =  ( Base `  K
)
2 cdlemefs32.l . 2  |-  .<_  =  ( le `  K )
3 cdlemefs32.j . 2  |-  .\/  =  ( join `  K )
4 cdlemefs32.m . 2  |-  ./\  =  ( meet `  K )
5 cdlemefs32.a . 2  |-  A  =  ( Atoms `  K )
6 cdlemefs32.h . 2  |-  H  =  ( LHyp `  K
)
7 breq1 4207 . 2  |-  ( s  =  R  ->  (
s  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  Q ) ) )
8 simp1 957 . . 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 )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
9 simp3l 985 . . . 4  |-  ( ( ( ( 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 )
10 simp3rl 1030 . . . 4  |-  ( ( ( ( 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  .<_  W )
119, 10jca 519 . . 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  /\  -.  s  .<_  W ) )
12 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
) )
13 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 )
14 cdlemefs32.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
15 cdlemefs32.d . . . 4  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
16 cdlemefs32.e . . . 4  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
17 cdlemefs32.i . . . 4  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
18 cdlemefs32.n . . . 4  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
191, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18cdlemefs27cl 31147 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  N  e.  B )
208, 11, 12, 13, 19syl13anc 1186 . 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 )
211, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18cdlemefs32snb 31149 . 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
)
22 cdleme29fs.o . 2  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
231, 2, 3, 4, 5, 6, 7, 20, 21, 22cdlemefrs32fva 31134 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   [_csb 3243   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LHypclh 30718
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722
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