Theorem dib11N 31643

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 3323	. 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 31642	. . . 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 31642	. . . . 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 1159	. . . 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 692	. . 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 981	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. HL )
11		hllat 29846	. . . . 5 |- ( K e. HL -> K e. Lat )
12	10, 11	syl 16	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. Lat )
13		simp2l 983	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> X e. B )
14		simp3l 985	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> Y e. B )
15	2, 3	latasymb 14438	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
16	12, 13, 14, 15	syl3anc 1184	. . 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 245	. 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 249	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 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   C_ wss 3280   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Latclat 14429   HLchlt 29833   LHypclh 30466   DIsoBcdib 31621

This theorem is referenced by:  dibf11N  31644

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-disoa 31512  df-dib 31622