Theorem diaord 36336

Description: The partial isomorphism A for a lattice K is order-preserving in the region under co-atom W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)

Hypotheses

Ref	Expression
dia11.b	|- B = ( Base ` K )
dia11.l	|- .<_ = ( le ` K )
dia11.h	|- H = ( LHyp ` K )
dia11.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

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

Proof of Theorem diaord

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dia11.b	. . . . 5 |- B = ( Base ` K )
2		dia11.l	. . . . 5 |- .<_ = ( le ` K )
3		dia11.h	. . . . 5 |- H = ( LHyp ` K )
4		eqid 2622	. . . . 5 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
5		eqid 2622	. . . . 5 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
6		dia11.i	. . . . 5 |- I = ( ( DIsoA ` K ) ` W )
7	1, 2, 3, 4, 5, 6	diaval 36321	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ X } )
8	7	3adant3 1081	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` X ) = { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ X } )
9	1, 2, 3, 4, 5, 6	diaval 36321	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` Y ) = { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ Y } )
10	9	3adant2 1080	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` Y ) = { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ Y } )
11	8, 10	sseq12d 3634	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ X } C_ { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ Y } ) )
12		eqid 2622	. . . 4 |- ( Atoms ` K ) = ( Atoms ` K )
13	1, 2, 12, 3, 4, 5	trlord 35857	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( X .<_ Y <-> A. f e. ( ( LTrn ` K ) ` W ) ( ( ( ( trL ` K ) ` W ) ` f ) .<_ X -> ( ( ( trL ` K ) ` W ) ` f ) .<_ Y ) ) )
14		ss2rab 3678	. . 3 |- ( { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ X } C_ { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ Y } <-> A. f e. ( ( LTrn ` K ) ` W ) ( ( ( ( trL ` K ) ` W ) ` f ) .<_ X -> ( ( ( trL ` K ) ` W ) ` f ) .<_ Y ) )
15	13, 14	syl6rbbr 279	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ X } C_ { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ Y } <-> X .<_ Y ) )
16	11, 15	bitrd 268	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> X .<_ Y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Atomscatm 34550   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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-iin 4523  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-1st 7168  df-2nd 7169  df-undef 7399  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-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-disoa 36318

This theorem is referenced by:  dia11N  36337  dia2dimlem10  36362  dibord  36448