Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemkfid3N Structured version   Unicode version

Theorem cdlemkfid3N 31784
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 992 . . 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 31779 . . 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 958 . . . . 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 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  e.  T
)
7 simp21r 1076 . . . . 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 1079 . . . . 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 994 . . . . 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 996 . . . . 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 31782 . . . . 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 1203 . . . 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 6098 . . 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 6099 . 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 982 . . 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 1080 . . 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 995 . . . 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 2689 . . 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 31780 . . 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 1203 . 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 2474 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   [_csb 3253   class class class wbr 4214    _I cid 4495   `'ccnv 4879    |` cres 4882    o. ccom 4884   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
  Copyright terms: Public domain W3C validator