Theorem dib11N 30723

Description: The isomorphism B for a lattice K is one-to-one in the region under co-atom W. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dib11.b	|- B = ( Base ` K )
dib11.l	|- .<_ = ( le ` K )
dib11.h	|- H = ( LHyp ` K )
dib11.i	|- I = ( ( DIsoB ` K ) ` W )

Assertion

Ref	Expression
dib11N	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) = ( I ` Y ) <-> X = Y ) )

Proof of Theorem dib11N

Step	Hyp	Ref	Expression
1		eqss 3194	. 2 |- ( ( I ` X ) = ( I ` Y ) <-> ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) )
2		dib11.b	. . . . 5 |- B = ( Base ` K )
3		dib11.l	. . . . 5 |- .<_ = ( le ` K )
4		dib11.h	. . . . 5 |- H = ( LHyp ` K )
5		dib11.i	. . . . 5 |- I = ( ( DIsoB ` K ) ` W )
6	2, 3, 4, 5	dibord 30722	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> X .<_ Y ) )
7	2, 3, 4, 5	dibord 30722	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( Y e. B /\ Y .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( I ` Y ) C_ ( I ` X ) <-> Y .<_ X ) )
8	7	3com23 1157	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` Y ) C_ ( I ` X ) <-> Y .<_ X ) )
9	6, 8	anbi12d 691	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) <-> ( X .<_ Y /\ Y .<_ X ) ) )
10		simp1l 979	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. HL )
11		hllat 28926	. . . . 5 |- ( K e. HL -> K e. Lat )
12	10, 11	syl 15	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. Lat )
13		simp2l 981	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> X e. B )
14		simp3l 983	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> Y e. B )
15	2, 3	latasymb 14160	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
16	12, 13, 14, 15	syl3anc 1182	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
17	9, 16	bitrd 244	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) <-> X = Y ) )
18	1, 17	syl5bb 248	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) = ( I ` Y ) <-> X = Y ) )

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1623   e. wcel 1684   C_ wss 3152   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Latclat 14151   HLchlt 28913   LHypclh 29546   DIsoBcdib 30701

This theorem is referenced by:  dibf11N  30724

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-llines 29060  df-lplanes 29061  df-lvols 29062  df-lines 29063  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550  df-laut 29551  df-ldil 29666  df-ltrn 29667  df-trl 29721  df-disoa 30592  df-dib 30702