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Theorem cdlemg28 31186
Description: Part of proof of Lemma G of [Crawley] p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 31187 (and maybe leading up to this too)? (Contributed by NM, 29-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
cdlemg33.o  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
Assertion
Ref Expression
cdlemg28  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
Distinct variable groups:    z, A    z, F    z, H    z,  .\/    z, K    z,  .<_    z, N    z, P    z, Q    z, R    z, T    z, W    z, v    z, G   
z, O
Allowed substitution hints:    A( v)    P( v)    Q( v)    R( v)    T( v)    F( v)    G( v)    H( v)    .\/ ( v)    K( v)   
.<_ ( v)    ./\ ( z, v)    N( v)    O( v)    W( v)

Proof of Theorem cdlemg28
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp21 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
4 simp22 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( z  e.  A  /\  -.  z  .<_  W ) )
5 simp23l 1078 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  F  e.  T
)
6 simp23r 1079 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  G  e.  T
)
7 simp32 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) )
8 simp313 1106 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  z  .<_  ( P 
.\/  v ) )
9 simp33 995 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) )
10 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
11 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
12 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
13 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
14 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
15 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
16 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
1710, 11, 12, 13, 14, 15, 16cdlemg28a 31175 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
)  /\  z  .<_  ( P  .\/  v )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( z 
.\/  ( F `  ( G `  z ) ) )  ./\  W
) )
181, 2, 3, 4, 5, 6, 7, 8, 9, 17syl333anc 1216 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( z  .\/  ( F `
 ( G `  z ) ) ) 
./\  W ) )
19 cdlemg31.n . . 3  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
20 cdlemg33.o . . 3  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
2110, 11, 12, 13, 14, 15, 16, 19, 20cdlemg28b 31185 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
)  =  ( ( z  .\/  ( F `
 ( G `  z ) ) ) 
./\  W ) )
2218, 21eqtr4d 2439 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  cdlemg29  31187
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-3or 937  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-map 6979  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-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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