Theorem trl0 35457

Description: If an atom not under the fiducial co-atom W equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)

Hypotheses

Ref	Expression
trl0.l	|- .<_ = ( le ` K )
trl0.z	|- .0. = ( 0. ` K )
trl0.a	|- A = ( Atoms ` K )
trl0.h	|- H = ( LHyp ` K )
trl0.t	|- T = ( ( LTrn ` K ) ` W )
trl0.r	|- R = ( ( trL ` K ) ` W )

Assertion

Ref	Expression
trl0	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = .0. )

Proof of Theorem trl0

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( K e. HL /\ W e. H ) )
2		simp3l 1089	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> F e. T )
3		simp2 1062	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P e. A /\ -. P .<_ W ) )
4		trl0.l	. . . 4 |- .<_ = ( le ` K )
5		eqid 2622	. . . 4 |- ( join ` K ) = ( join ` K )
6		eqid 2622	. . . 4 |- ( meet ` K ) = ( meet ` K )
7		trl0.a	. . . 4 |- A = ( Atoms ` K )
8		trl0.h	. . . 4 |- H = ( LHyp ` K )
9		trl0.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
10		trl0.r	. . . 4 |- R = ( ( trL ` K ) ` W )
11	4, 5, 6, 7, 8, 9, 10	trlval2 35450	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) )
12	1, 2, 3, 11	syl3anc 1326	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) )
13		simp3r 1090	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( F ` P ) = P )
14	13	oveq2d 6666	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) ( F ` P ) ) = ( P ( join ` K ) P ) )
15		simp1l 1085	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> K e. HL )
16		simp2l 1087	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> P e. A )
17	5, 7	hlatjidm 34655	. . . . 5 |- ( ( K e. HL /\ P e. A ) -> ( P ( join ` K ) P ) = P )
18	15, 16, 17	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) P ) = P )
19	14, 18	eqtrd 2656	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) ( F ` P ) ) = P )
20	19	oveq1d 6665	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) = ( P ( meet ` K ) W ) )
21		trl0.z	. . . 4 |- .0. = ( 0. ` K )
22	4, 6, 21, 7, 8	lhpmat 35316	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ( meet ` K ) W ) = .0. )
23	1, 3, 22	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( meet ` K ) W ) = .0. )
24	12, 20, 23	3eqtrd 2660	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   meetcmee 16945   0.cp0 17037   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-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-map 7859  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-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  trlator0  35458  ltrnnidn  35461  trlid0  35463  trlnidatb  35464  trlnle  35473  trlval3  35474  trlval4  35475  cdlemc6  35483  cdlemg31d  35988