Theorem dihord6b 36549

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Hypotheses

Ref	Expression
dihord3.b	|- B = ( Base ` K )
dihord3.l	|- .<_ = ( le ` K )
dihord3.h	|- H = ( LHyp ` K )
dihord3.i	|- I = ( ( DIsoH ` K ) ` W )

Assertion

Ref	Expression
dihord6b	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) /\ X .<_ Y ) -> ( I ` X ) C_ ( I ` Y ) )

Proof of Theorem dihord6b

Step	Hyp	Ref	Expression
1		simp2r 1088	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> -. X .<_ W )
2		simp3r 1090	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> Y .<_ W )
3		simp1l 1085	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. HL )
4		hllat 34650	. . . . . . 7 |- ( K e. HL -> K e. Lat )
5	3, 4	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. Lat )
6		simp2l 1087	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> X e. B )
7		simp3l 1089	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> Y e. B )
8		simp1r 1086	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> W e. H )
9		dihord3.b	. . . . . . . 8 |- B = ( Base ` K )
10		dihord3.h	. . . . . . . 8 |- H = ( LHyp ` K )
11	9, 10	lhpbase 35284	. . . . . . 7 |- ( W e. H -> W e. B )
12	8, 11	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> W e. B )
13		dihord3.l	. . . . . . 7 |- .<_ = ( le ` K )
14	9, 13	lattr 17056	. . . . . 6 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ W e. B ) ) -> ( ( X .<_ Y /\ Y .<_ W ) -> X .<_ W ) )
15	5, 6, 7, 12, 14	syl13anc 1328	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( X .<_ Y /\ Y .<_ W ) -> X .<_ W ) )
16	2, 15	mpan2d 710	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( X .<_ Y -> X .<_ W ) )
17	1, 16	mtod 189	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> -. X .<_ Y )
18	17	pm2.21d 118	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( X .<_ Y -> ( I ` X ) C_ ( I ` Y ) ) )
19	18	imp 445	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) /\ X .<_ Y ) -> ( I ` X ) C_ ( I ` Y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Latclat 17045   HLchlt 34637   LHypclh 35270   DIsoHcdih 36517

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-poset 16946  df-lat 17046  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274

This theorem is referenced by:  dihord  36553