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Theorem cdleme4gfv 29963
Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: Can this replace uses of cdleme32a 29897? 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 29949 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 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 ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp13 992 . 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 991 . 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 986 . . 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 2530 . 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 987 . 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 962 . 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 29955 . 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 1209 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 set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   A.wral 2544   [_csb 3082   ifcif 3566   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   iota_crio 6290   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440
This theorem is referenced by:  cdleme48d  29991
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444
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