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Theorem cdlemkfid3N 31407
Description: TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
Assertion
Ref Expression
cdlemkfid3N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  [_ G  /  g ]_ Y  =  ( G `  P )
)
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b    g, G
Allowed substitution hints:    A( g, b)    B( b)    P( b)    R( b)    T( b)    F( g, b)    G( b)    H( g, b)    .\/ ( b)    K( g,
b)    .<_ ( g, b)    ./\ ( b)    N( g, b)    W( g, b)    Y( g, b)    Z( b)

Proof of Theorem cdlemkfid3N
StepHypRef Expression
1 simp22 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  e.  T
)
2 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
32cdlemk41 31402 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
41, 3syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  [_ G  /  g ]_ Y  =  (
( P  .\/  ( R `  G )
)  ./\  ( Z  .\/  ( R `  ( G  o.  `' b
) ) ) ) )
5 simp1 957 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N ) )
6 simp21l 1074 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T
)
7 simp21r 1075 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
8 simp23l 1078 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  b  e.  T
)
9 simp31 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  b )  =/=  ( R `  F )
)
10 simp33 995 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
11 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
12 cdlemk5.l . . . . . 6  |-  .<_  =  ( le `  K )
13 cdlemk5.j . . . . . 6  |-  .\/  =  ( join `  K )
14 cdlemk5.m . . . . . 6  |-  ./\  =  ( meet `  K )
15 cdlemk5.a . . . . . 6  |-  A  =  ( Atoms `  K )
16 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
17 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
18 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
19 cdlemk5.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
2011, 12, 13, 14, 15, 16, 17, 18, 19cdlemkfid2N 31405 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  Z  =  ( b `  P ) )
215, 6, 7, 8, 9, 10, 20syl132anc 1202 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  Z  =  ( b `  P ) )
2221oveq1d 6055 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( Z  .\/  ( R `  ( G  o.  `' b ) ) )  =  ( ( b `  P
)  .\/  ( R `  ( G  o.  `' b ) ) ) )
2322oveq2d 6056 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
b `  P )  .\/  ( R `  ( G  o.  `' b
) ) ) ) )
24 simp1l 981 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
25 simp23r 1079 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  b  =/=  (  _I  |`  B ) )
26 simp32 994 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  b )  =/=  ( R `  G )
)
2726necomd 2650 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  G )  =/=  ( R `  b )
)
2811, 12, 13, 14, 15, 16, 17, 18cdlemkfid1N 31403 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B )  /\  G  e.  T
)  /\  ( ( R `  G )  =/=  ( R `  b
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( b `  P )  .\/  ( R `  ( G  o.  `' b ) ) ) )  =  ( G `  P ) )
2924, 8, 25, 1, 27, 10, 28syl132anc 1202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( b `  P )  .\/  ( R `  ( G  o.  `' b ) ) ) )  =  ( G `  P ) )
304, 23, 293eqtrd 2440 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  [_ G  /  g ]_ Y  =  ( G `  P )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   [_csb 3211   class class class wbr 4172    _I cid 4453   `'ccnv 4836    |` cres 4839    o. ccom 4841   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
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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