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Theorem cdleme42ke 35773
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 1112 . . . . 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 1068 . . . . . . 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 35727 . . . . . . 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 487 . . . . . 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 475 . . . . 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 34655 . . . . 5 |- ( ( K e. HL /\ ( F ` R ) e. A ) -> ( ( F ` R ) .\/ ( F ` R ) ) = ( F ` R ) )
211, 19, 20syl2anc 693 . . . 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 6191 . . . . 5 |- ( R = S -> ( F ` R ) = ( F ` S ) )
2322oveq2d 6666 . . . 4 |- ( R = S -> ( ( F ` R ) .\/ ( F ` R ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )
2421, 23sylan9req 2677 . . 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 1120 . . . . . . . . 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 34655 . . . . . . . . 9 |- ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
271, 25, 26syl2anc 693 . . . . . . . 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 6665 . . . . . . 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 1064 . . . . . . . 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 2622 . . . . . . . . 9 |- ( 0. ` K ) = ( 0. ` K )
314, 6, 30, 7, 8lhpmat 35316 . . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R ./\ W ) = ( 0. ` K ) )
3229, 2, 31syl2anc 693 . . . . . . 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 2656 . . . . . 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 6666 . . . . 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 34648 . . . . . . 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 34576 . . . . . . 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 34512 . . . . . 6 |- ( ( K e. OL /\ ( F ` R ) e. B ) -> ( ( F ` R ) .\/ ( 0. ` K ) ) = ( F ` R ) )
4036, 38, 39syl2anc 693 . . . . 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 2656 . . . 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 6658 . . . . . . 7 |- ( R = S -> ( R .\/ R ) = ( R .\/ S ) )
4342oveq1d 6665 . . . . . 6 |- ( R = S -> ( ( R .\/ R ) ./\ W ) = ( ( R .\/ S ) ./\ W ) )
44 cdleme34e.v . . . . . 6 |- V = ( ( R .\/ S ) ./\ W )
4543, 44syl6eqr 2674 . . . . 5 |- ( R = S -> ( ( R .\/ R ) ./\ W ) = V )
4645oveq2d 6666 . . . 4 |- ( R = S -> ( ( F ` R ) .\/ ( ( R .\/ R ) ./\ W ) ) = ( ( F ` R ) .\/ V ) )
4741, 46sylan9req 2677 . . 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 2658 . 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 35772 . . 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 1265 . 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 2881 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   0.cp0 17037   OLcol 34461   Atomscatm 34550   HLchlt 34637   LHypclh 35270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274
This theorem is referenced by:  cdleme42keg  35774  cdleme42mN  35775  cdlemeg46fjv  35811
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