Theorem ltrncnvel 35428

Description: The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)

Hypotheses

Ref	Expression
ltrnel.l	|- .<_ = ( le ` K )
ltrnel.a	|- A = ( Atoms ` K )
ltrnel.h	|- H = ( LHyp ` K )
ltrnel.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
ltrncnvel	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( `' F ` P ) e. A /\ -. ( `' F ` P ) .<_ W ) )

Proof of Theorem ltrncnvel

Step	Hyp	Ref	Expression
1		ltrnel.l	. . . 4 |- .<_ = ( le ` K )
2		ltrnel.a	. . . 4 |- A = ( Atoms ` K )
3		ltrnel.h	. . . 4 |- H = ( LHyp ` K )
4		ltrnel.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
5	1, 2, 3, 4	ltrncnvat 35427	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( `' F ` P ) e. A )
6	5	3adant3r 1323	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( `' F ` P ) e. A )
7		simp3r 1090	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> -. P .<_ W )
8		simp1 1061	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
9		simp2 1062	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
10		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
11	10, 2	atbase 34576	. . . . . 6 |- ( ( `' F ` P ) e. A -> ( `' F ` P ) e. ( Base ` K ) )
12	6, 11	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( `' F ` P ) e. ( Base ` K ) )
13		simp1r 1086	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. H )
14	10, 3	lhpbase 35284	. . . . . 6 |- ( W e. H -> W e. ( Base ` K ) )
15	13, 14	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. ( Base ` K ) )
16	10, 1, 3, 4	ltrnle 35415	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( `' F ` P ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( `' F ` P ) .<_ W <-> ( F ` ( `' F ` P ) ) .<_ ( F ` W ) ) )
17	8, 9, 12, 15, 16	syl112anc 1330	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( `' F ` P ) .<_ W <-> ( F ` ( `' F ` P ) ) .<_ ( F ` W ) ) )
18	10, 3, 4	ltrn1o 35410	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
19	18	3adant3 1081	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
20		simp3l 1089	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
21	10, 2	atbase 34576	. . . . . . 7 |- ( P e. A -> P e. ( Base ` K ) )
22	20, 21	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. ( Base ` K ) )
23		f1ocnvfv2 6533	. . . . . 6 |- ( ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( F ` ( `' F ` P ) ) = P )
24	19, 22, 23	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` ( `' F ` P ) ) = P )
25		simp1l 1085	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
26		hllat 34650	. . . . . . . 8 |- ( K e. HL -> K e. Lat )
27	25, 26	syl 17	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. Lat )
28	10, 1	latref 17053	. . . . . . 7 |- ( ( K e. Lat /\ W e. ( Base ` K ) ) -> W .<_ W )
29	27, 15, 28	syl2anc 693	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W .<_ W )
30	10, 1, 3, 4	ltrnval1 35420	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( W e. ( Base ` K ) /\ W .<_ W ) ) -> ( F ` W ) = W )
31	8, 9, 15, 29, 30	syl112anc 1330	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` W ) = W )
32	24, 31	breq12d 4666	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` ( `' F ` P ) ) .<_ ( F ` W ) <-> P .<_ W ) )
33	17, 32	bitrd 268	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( `' F ` P ) .<_ W <-> P .<_ W ) )
34	7, 33	mtbird 315	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> -. ( `' F ` P ) .<_ W )
35	6, 34	jca 554	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( `' F ` P ) e. A /\ -. ( `' F ` P ) .<_ W ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   lecple 15948   Latclat 17045   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387

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-glb 16975  df-p0 17039  df-lat 17046  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

This theorem is referenced by:  ltrncnv  35432  cdlemg17h  35956