Theorem dia0 36341

Description: The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)

Hypotheses

Ref	Expression
dia0.b	|- B = ( Base ` K )
dia0.z	|- .0. = ( 0. ` K )
dia0.h	|- H = ( LHyp ` K )
dia0.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

Ref	Expression
dia0	|- ( ( K e. HL /\ W e. H ) -> ( I ` .0. ) = { ( _I |` B ) } )

Proof of Theorem dia0

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( K e. HL /\ W e. H ) )
2		hlatl 34647	. . . . 5 |- ( K e. HL -> K e. AtLat )
3		dia0.b	. . . . . 6 |- B = ( Base ` K )
4		dia0.z	. . . . . 6 |- .0. = ( 0. ` K )
5	3, 4	atl0cl 34590	. . . . 5 |- ( K e. AtLat -> .0. e. B )
6	2, 5	syl 17	. . . 4 |- ( K e. HL -> .0. e. B )
7	6	adantr 481	. . 3 |- ( ( K e. HL /\ W e. H ) -> .0. e. B )
8		dia0.h	. . . . 5 |- H = ( LHyp ` K )
9	3, 8	lhpbase 35284	. . . 4 |- ( W e. H -> W e. B )
10		eqid 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
11	3, 10, 4	atl0le 34591	. . . 4 |- ( ( K e. AtLat /\ W e. B ) -> .0. ( le ` K ) W )
12	2, 9, 11	syl2an 494	. . 3 |- ( ( K e. HL /\ W e. H ) -> .0. ( le ` K ) W )
13		eqid 2622	. . . 4 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
14		eqid 2622	. . . 4 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
15		dia0.i	. . . 4 |- I = ( ( DIsoA ` K ) ` W )
16	3, 10, 8, 13, 14, 15	diaval 36321	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( .0. e. B /\ .0. ( le ` K ) W ) ) -> ( I ` .0. ) = { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) .0. } )
17	1, 7, 12, 16	syl12anc 1324	. 2 |- ( ( K e. HL /\ W e. H ) -> ( I ` .0. ) = { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) .0. } )
18	2	ad2antrr 762	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) ) -> K e. AtLat )
19	3, 8, 13, 14	trlcl 35451	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( trL ` K ) ` W ) ` f ) e. B )
20	3, 10, 4	atlle0 34592	. . . . 5 |- ( ( K e. AtLat /\ ( ( ( trL ` K ) ` W ) ` f ) e. B ) -> ( ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) .0. <-> ( ( ( trL ` K ) ` W ) ` f ) = .0. ) )
21	18, 19, 20	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) .0. <-> ( ( ( trL ` K ) ` W ) ` f ) = .0. ) )
22	3, 4, 8, 13, 14	trlid0b 35465	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) ) -> ( f = ( _I |` B ) <-> ( ( ( trL ` K ) ` W ) ` f ) = .0. ) )
23	21, 22	bitr4d 271	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) .0. <-> f = ( _I |` B ) ) )
24	23	rabbidva 3188	. 2 |- ( ( K e. HL /\ W e. H ) -> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) .0. } = { f e. ( ( LTrn ` K ) ` W ) | f = ( _I |` B ) } )
25	3, 8, 13	idltrn 35436	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) )
26		rabsn 4256	. . 3 |- ( ( _I |` B ) e. ( ( LTrn ` K ) ` W ) -> { f e. ( ( LTrn ` K ) ` W ) | f = ( _I |` B ) } = { ( _I |` B ) } )
27	25, 26	syl 17	. 2 |- ( ( K e. HL /\ W e. H ) -> { f e. ( ( LTrn ` K ) ` W ) | f = ( _I |` B ) } = { ( _I |` B ) } )
28	17, 24, 27	3eqtrd 2660	1 |- ( ( K e. HL /\ W e. H ) -> ( I ` .0. ) = { ( _I |` B ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   {csn 4177   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   0.cp0 17037   AtLatcal 34551   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   DIsoAcdia 36317

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  df-disoa 36318

This theorem is referenced by:  dib0  36453