Theorem trlnidatb 35464

Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 35460? Why do both this and ltrnideq 35462 need trlnidat 35460? (Contributed by NM, 4-Jun-2013.)

Hypotheses

Ref	Expression
trlnidatb.b	|- B = ( Base ` K )
trlnidatb.a	|- A = ( Atoms ` K )
trlnidatb.h	|- H = ( LHyp ` K )
trlnidatb.t	|- T = ( ( LTrn ` K ) ` W )
trlnidatb.r	|- R = ( ( trL ` K ) ` W )

Assertion

Ref	Expression
trlnidatb	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) <-> ( R ` F ) e. A ) )

Proof of Theorem trlnidatb

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlnidatb.b	. . . 4 |- B = ( Base ` K )
2		trlnidatb.a	. . . 4 |- A = ( Atoms ` K )
3		trlnidatb.h	. . . 4 |- H = ( LHyp ` K )
4		trlnidatb.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
5		trlnidatb.r	. . . 4 |- R = ( ( trL ` K ) ` W )
6	1, 2, 3, 4, 5	trlnidat 35460	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. A )
7	6	3expia 1267	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) -> ( R ` F ) e. A ) )
8		eqid 2622	. . . . . 6 |- ( le ` K ) = ( le ` K )
9	8, 2, 3	lhpexnle 35292	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> E. p e. A -. p ( le ` K ) W )
10	9	adantr 481	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> E. p e. A -. p ( le ` K ) W )
11	1, 8, 2, 3, 4	ltrnideq 35462	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( F = ( _I |` B ) <-> ( F ` p ) = p ) )
12	11	3expa 1265	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( F = ( _I |` B ) <-> ( F ` p ) = p ) )
13		simp1l 1085	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( K e. HL /\ W e. H ) )
14		simp2 1062	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( p e. A /\ -. p ( le ` K ) W ) )
15		simp1r 1086	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> F e. T )
16		simp3 1063	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( F ` p ) = p )
17		eqid 2622	. . . . . . . . 9 |- ( 0. ` K ) = ( 0. ` K )
18	8, 17, 2, 3, 4, 5	trl0 35457	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F e. T /\ ( F ` p ) = p ) ) -> ( R ` F ) = ( 0. ` K ) )
19	13, 14, 15, 16, 18	syl112anc 1330	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( R ` F ) = ( 0. ` K ) )
20	19	3expia 1267	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( F ` p ) = p -> ( R ` F ) = ( 0. ` K ) ) )
21		simplll 798	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> K e. HL )
22		hlatl 34647	. . . . . . . 8 |- ( K e. HL -> K e. AtLat )
23	17, 2	atn0 34595	. . . . . . . . 9 |- ( ( K e. AtLat /\ ( R ` F ) e. A ) -> ( R ` F ) =/= ( 0. ` K ) )
24	23	ex 450	. . . . . . . 8 |- ( K e. AtLat -> ( ( R ` F ) e. A -> ( R ` F ) =/= ( 0. ` K ) ) )
25	21, 22, 24	3syl 18	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( R ` F ) e. A -> ( R ` F ) =/= ( 0. ` K ) ) )
26	25	necon2bd 2810	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( R ` F ) = ( 0. ` K ) -> -. ( R ` F ) e. A ) )
27	20, 26	syld 47	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( F ` p ) = p -> -. ( R ` F ) e. A ) )
28	12, 27	sylbid 230	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( F = ( _I |` B ) -> -. ( R ` F ) e. A ) )
29	10, 28	rexlimddv 3035	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F = ( _I |` B ) -> -. ( R ` F ) e. A ) )
30	29	necon2ad 2809	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( R ` F ) e. A -> F =/= ( _I |` B ) ) )
31	7, 30	impbid 202	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) <-> ( R ` F ) e. A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   0.cp0 17037   Atomscatm 34550   AtLatcal 34551   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-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-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  trlid0b  35465  cdlemfnid  35852  trlconid  36013  dih1dimb2  36530