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Theorem cdlemeg47b 34152
Description: TODO FIX COMMENT (Contributed by NM, 1-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
cdlemeg47b  |-  ( ( ( ( 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
) )  ->  ( G `  S )  =  [_ S  /  v ]_ N )
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
Allowed substitution hints:    G( v, u, a, b, c)    N( v)    O( v, u)

Proof of Theorem cdlemeg47b
StepHypRef Expression
1 cdlemef47.j . . 3  |-  .\/  =  ( join `  K )
2 cdlemef47.a . . 3  |-  A  =  ( Atoms `  K )
31, 2cdleme46f2g2 34137 . 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
) )  ->  (
( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  =/=  P  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( Q  .\/  P
) ) )
4 cdlemef47.b . . 3  |-  B  =  ( Base `  K
)
5 cdlemef47.l . . 3  |-  .<_  =  ( le `  K )
6 cdlemef47.m . . 3  |-  ./\  =  ( meet `  K )
7 cdlemef47.h . . 3  |-  H  =  ( LHyp `  K
)
8 cdlemef47.v . . 3  |-  V  =  ( ( Q  .\/  P )  ./\  W )
9 cdlemef47.n . . 3  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
10 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 ) )
114, 5, 1, 6, 2, 7, 8, 9, 10cdlemefr45 34071 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  =/=  P  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( Q  .\/  P
) )  ->  ( G `  S )  =  [_ S  /  v ]_ N )
123, 11syl 16 1  |-  ( ( ( ( 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
) )  ->  ( G `  S )  =  [_ S  /  v ]_ N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   [_csb 3288   ifcif 3791   class class class wbr 4292    e. cmpt 4350   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   Atomscatm 32908   HLchlt 32995   LHypclh 33628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-lines 33145  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632
This theorem is referenced by:  cdlemeg47rv2  34154  cdlemeg46bOLDN  34156  cdlemeg46c  34157  cdlemeg46rjgN  34166
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