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Theorem cdleme4gfv 34460
Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: Can this replace uses of cdleme32a 34394? TODO: Can this be used to help prove the  R or  S case where  X is an atom? TODO: Would an antecedent transformer like cdleme46f2g2 34446 help? (Contributed by NM, 8-Apr-2013.)
Hypotheses
Ref Expression
cdlemef47.b  |-  B  =  ( Base `  K
)
cdlemef47.l  |-  .<_  =  ( le `  K )
cdlemef47.j  |-  .\/  =  ( join `  K )
cdlemef47.m  |-  ./\  =  ( meet `  K )
cdlemef47.a  |-  A  =  ( Atoms `  K )
cdlemef47.h  |-  H  =  ( LHyp `  K
)
cdlemef47.v  |-  V  =  ( ( Q  .\/  P )  ./\  W )
cdlemef47.n  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
cdlemefs47.o  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
cdlemef47.g  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B  A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
Assertion
Ref Expression
cdleme4gfv  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  X )  =  ( ( G `  S
)  .\/  ( X  ./\ 
W ) ) )
Distinct variable groups:    a, b,
c, u, v, A    B, a, b, c, u, v    H, a, b, c, u, v    .\/ , a,
b, c, u, v    K, a, b, c, u, v    .<_ , a, b, c, u, v    ./\ , a,
b, c, u, v    N, a, b, c, u    O, a, b, c    P, a, b, c, u, v    Q, a, b, c, u, v    S, a, b, c, u, v    V, a, b, c, u, v    W, a, b, c, u, v    X, a, c, u, v
Allowed substitution hints:    G( v, u, a, b, c)    N( v)    O( v, u)    X( b)

Proof of Theorem cdleme4gfv
StepHypRef Expression
1 simp11 1018 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp13 1020 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3 simp12 1019 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp2l 1014 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  P  =/=  Q )
54necomd 2719 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  Q  =/=  P )
6 simp2r 1015 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
7 simp3 990 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )
8 cdlemef47.b . . 3  |-  B  =  ( Base `  K
)
9 cdlemef47.l . . 3  |-  .<_  =  ( le `  K )
10 cdlemef47.j . . 3  |-  .\/  =  ( join `  K )
11 cdlemef47.m . . 3  |-  ./\  =  ( meet `  K )
12 cdlemef47.a . . 3  |-  A  =  ( Atoms `  K )
13 cdlemef47.h . . 3  |-  H  =  ( LHyp `  K
)
14 cdlemef47.v . . 3  |-  V  =  ( ( Q  .\/  P )  ./\  W )
15 cdlemef47.n . . 3  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
16 cdlemefs47.o . . 3  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
17 cdlemef47.g . . 3  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B  A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
188, 9, 10, 11, 12, 13, 14, 15, 16, 17cdleme48fv 34452 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  =/=  P  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  X )  =  ( ( G `  S
)  .\/  ( X  ./\ 
W ) ) )
191, 2, 3, 5, 6, 7, 18syl321anc 1241 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  X )  =  ( ( G `  S
)  .\/  ( X  ./\ 
W ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   [_csb 3389   ifcif 3892   class class class wbr 4393    |-> cmpt 4451   ` cfv 5519   iota_crio 6153  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   meetcmee 15226   Atomscatm 33217   HLchlt 33304   LHypclh 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-riotaBAD 32913
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-undef 6895  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941
This theorem is referenced by:  cdleme48d  34488
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