Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme42ke Structured version   Visualization version   Unicode version
Theorem cdleme42ke 34064
Description: Part of proof of Lemma E in [Crawley] p. 113. Remove R =/= S condition. TODO: FIX COMMENT. (Contributed by NM, 2-Apr-2013.)
Hypotheses
Ref Expression
cdleme41.b |- B = ( Base ` K )
cdleme41.l |- .<_ = ( le ` K )
cdleme41.j |- .\/ = ( join ` K )
cdleme41.m |- ./\ = ( meet ` K )
cdleme41.a |- A = ( Atoms ` K )
cdleme41.h |- H = ( LHyp ` K )
cdleme41.u |- U = ( ( P .\/ Q ) ./\ W )
cdleme41.d |- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme41.e |- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme41.g |- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme41.i |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
cdleme41.n |- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme41.o |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme41.f |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdleme34e.v |- V = ( ( R .\/ S ) ./\ W )
Assertion
Ref Expression
cdleme42ke |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) )
Distinct variable groups:   A, s   .\/ , s   .<_ , s   ./\ , s   P, s   Q, s   R, s   S, s   U, s   W, s   y, t, A, s   B, s, t, y   y, D   y, G   E, s, y   H, s, t, y   t, .\/ , y   K, s, t, y   t, .<_ , y   t, ./\ , y   t, P, y   t, Q, y   t, R, y   t, S, y   t, U, y   t, W, y   x, z, A   x, B, z   z, E, s   z, H   x, .\/ , z   z, K   x, .<_ , z   x, ./\ , z   x, N, z   x, P, z   x, Q, z   x, R, z   x, S, z   x, U, z   x, W, z, s, t, y   V, s, t, x, z
Allowed substitution hints:   D( x, z, t, s)   E( x, t)   F( x, y, z, t, s)   G( x, z, t, s)   H( x)   I( x, y, z, t, s)   K( x)   N( y, t, s)   O( x, y, z, t, s)   V( y)
Proof of Theorem cdleme42ke
StepHypRef Expression
1 simpl1l 1060 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> K e. HL )
2 simpr2 1016 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
3 cdleme41.b . . . . . . . 8 |- B = ( Base ` K )
4 cdleme41.l . . . . . . . 8 |- .<_ = ( le ` K )
5 cdleme41.j . . . . . . . 8 |- .\/ = ( join ` K )
6 cdleme41.m . . . . . . . 8 |- ./\ = ( meet ` K )
7 cdleme41.a . . . . . . . 8 |- A = ( Atoms ` K )
8 cdleme41.h . . . . . . . 8 |- H = ( LHyp ` K )
9 cdleme41.u . . . . . . . 8 |- U = ( ( P .\/ Q ) ./\ W )
10 cdleme41.d . . . . . . . 8 |- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
11 cdleme41.e . . . . . . . 8 |- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
12 cdleme41.g . . . . . . . 8 |- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
13 cdleme41.i . . . . . . . 8 |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
14 cdleme41.n . . . . . . . 8 |- N = if ( s .<_ ( P .\/ Q ) , I , D )
15 cdleme41.o . . . . . . . 8 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
16 cdleme41.f . . . . . . . 8 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
173, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16cdleme32fvaw 34018 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( F ` R ) e. A /\ -. ( F ` R ) .<_ W ) )
182, 17syldan 473 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( F ` R ) e. A /\ -. ( F ` R ) .<_ W ) )
1918simpld 461 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( F ` R ) e. A )
205, 7hlatjidm 32946 . . . . 5 |- ( ( K e. HL /\ ( F ` R ) e. A ) -> ( ( F ` R ) .\/ ( F ` R ) ) = ( F ` R ) )
211, 19, 20syl2anc 667 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( F ` R ) .\/ ( F ` R ) ) = ( F ` R ) )
22 fveq2 5870 . . . . 5 |- ( R = S -> ( F ` R ) = ( F ` S ) )
2322oveq2d 6311 . . . 4 |- ( R = S -> ( ( F ` R ) .\/ ( F ` R ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )
2421, 23sylan9req 2508 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ R = S ) -> ( F ` R ) = ( ( F ` R ) .\/ ( F ` S ) ) )
25 simpr2l 1068 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> R e. A )
265, 7hlatjidm 32946 . . . . . . . . 9 |- ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
271, 25, 26syl2anc 667 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( R .\/ R ) = R )
2827oveq1d 6310 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( R .\/ R ) ./\ W ) = ( R ./\ W ) )
29 simpl1 1012 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
30 eqid 2453 . . . . . . . . 9 |- ( 0. ` K ) = ( 0. ` K )
314, 6, 30, 7, 8lhpmat 33607 . . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R ./\ W ) = ( 0. ` K ) )
3229, 2, 31syl2anc 667 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( R ./\ W ) = ( 0. ` K ) )
3328, 32eqtrd 2487 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( R .\/ R ) ./\ W ) = ( 0. ` K ) )
3433oveq2d 6311 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( F ` R ) .\/ ( ( R .\/ R ) ./\ W ) ) = ( ( F ` R ) .\/ ( 0. ` K ) ) )
35 hlol 32939 . . . . . . 7 |- ( K e. HL -> K e. OL )
361, 35syl 17 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> K e. OL )
373, 7atbase 32867 . . . . . . 7 |- ( ( F ` R ) e. A -> ( F ` R ) e. B )
3819, 37syl 17 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( F ` R ) e. B )
393, 5, 30olj01 32803 . . . . . 6 |- ( ( K e. OL /\ ( F ` R ) e. B ) -> ( ( F ` R ) .\/ ( 0. ` K ) ) = ( F ` R ) )
4036, 38, 39syl2anc 667 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( F ` R ) .\/ ( 0. ` K ) ) = ( F ` R ) )
4134, 40eqtrd 2487 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( F ` R ) .\/ ( ( R .\/ R ) ./\ W ) ) = ( F ` R ) )
42 oveq2 6303 . . . . . . 7 |- ( R = S -> ( R .\/ R ) = ( R .\/ S ) )
4342oveq1d 6310 . . . . . 6 |- ( R = S -> ( ( R .\/ R ) ./\ W ) = ( ( R .\/ S ) ./\ W ) )
44 cdleme34e.v . . . . . 6 |- V = ( ( R .\/ S ) ./\ W )
4543, 44syl6eqr 2505 . . . . 5 |- ( R = S -> ( ( R .\/ R ) ./\ W ) = V )
4645oveq2d 6311 . . . 4 |- ( R = S -> ( ( F ` R ) .\/ ( ( R .\/ R ) ./\ W ) ) = ( ( F ` R ) .\/ V ) )
4741, 46sylan9req 2508 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ R = S ) -> ( F ` R ) = ( ( F ` R ) .\/ V ) )
4824, 47eqtr3d 2489 . 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ R = S ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) )
493, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44cdleme42k 34063 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ R =/= S ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) )
50493expa 1209 . 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ R =/= S ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) )
5148, 50pm2.61dane 2713 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   A.wral 2739   ifcif 3883   class class class wbr 4405   |-> cmpt 4464   ` cfv 5585   iota_crio 6256  (class class class)co 6295   Basecbs 15133   lecple 15209   joincjn 16201   meetcmee 16202   0.cp0 16295   OLcol 32752   Atomscatm 32841   HLchlt 32928   LHypclh 33561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-riotaBAD 32537
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  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 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-undef 7025  df-preset 16185  df-poset 16203  df-plt 16216  df-lub 16232  df-glb 16233  df-join 16234  df-meet 16235  df-p0 16297  df-p1 16298  df-lat 16304  df-clat 16366  df-oposet 32754  df-ol 32756  df-oml 32757  df-covers 32844  df-ats 32845  df-atl 32876  df-cvlat 32900  df-hlat 32929  df-llines 33075  df-lplanes 33076  df-lvols 33077  df-lines 33078  df-psubsp 33080  df-pmap 33081  df-padd 33373  df-lhyp 33565
This theorem is referenced by:  cdleme42keg  34065  cdleme42mN  34066  cdlemeg46fjv  34102
  Copyright terms: Public domain W3C validator