Theorem dihmeetbclemN 36593

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem dihmeetbclemN

Step	Hyp	Ref	Expression
1		simp3 1063	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( X ./\ Y ) .<_ W )
2		simp1l 1085	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> K e. HL )
3		hllat 34650	. . . . . . . 8 |- ( K e. HL -> K e. Lat )
4	2, 3	syl 17	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> K e. Lat )
5		simp2l 1087	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> X e. B )
6		simp2r 1088	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> Y e. B )
7		dihmeetc.b	. . . . . . . . 9 |- B = ( Base ` K )
8		dihmeetc.m	. . . . . . . . 9 |- ./\ = ( meet ` K )
9	7, 8	latmcl 17052	. . . . . . . 8 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
10	4, 5, 6, 9	syl3anc 1326	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( X ./\ Y ) e. B )
11		simp1r 1086	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> W e. H )
12		dihmeetc.h	. . . . . . . . 9 |- H = ( LHyp ` K )
13	7, 12	lhpbase 35284	. . . . . . . 8 |- ( W e. H -> W e. B )
14	11, 13	syl 17	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> W e. B )
15		dihmeetc.l	. . . . . . . 8 |- .<_ = ( le ` K )
16	7, 15, 8	latleeqm1 17079	. . . . . . 7 |- ( ( K e. Lat /\ ( X ./\ Y ) e. B /\ W e. B ) -> ( ( X ./\ Y ) .<_ W <-> ( ( X ./\ Y ) ./\ W ) = ( X ./\ Y ) ) )
17	4, 10, 14, 16	syl3anc 1326	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( ( X ./\ Y ) .<_ W <-> ( ( X ./\ Y ) ./\ W ) = ( X ./\ Y ) ) )
18	1, 17	mpbid 222	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( ( X ./\ Y ) ./\ W ) = ( X ./\ Y ) )
19		hlol 34648	. . . . . . 7 |- ( K e. HL -> K e. OL )
20	2, 19	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> K e. OL )
21	7, 8	latmassOLD 34516	. . . . . 6 |- ( ( K e. OL /\ ( X e. B /\ Y e. B /\ W e. B ) ) -> ( ( X ./\ Y ) ./\ W ) = ( X ./\ ( Y ./\ W ) ) )
22	20, 5, 6, 14, 21	syl13anc 1328	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( ( X ./\ Y ) ./\ W ) = ( X ./\ ( Y ./\ W ) ) )
23	18, 22	eqtr3d 2658	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( X ./\ Y ) = ( X ./\ ( Y ./\ W ) ) )
24	23	fveq2d 6195	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( I ` ( X ./\ Y ) ) = ( I ` ( X ./\ ( Y ./\ W ) ) ) )
25		simp1 1061	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( K e. HL /\ W e. H ) )
26	7, 8	latmcl 17052	. . . . 5 |- ( ( K e. Lat /\ Y e. B /\ W e. B ) -> ( Y ./\ W ) e. B )
27	4, 6, 14, 26	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( Y ./\ W ) e. B )
28	7, 15, 8	latmle2 17077	. . . . 5 |- ( ( K e. Lat /\ Y e. B /\ W e. B ) -> ( Y ./\ W ) .<_ W )
29	4, 6, 14, 28	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( Y ./\ W ) .<_ W )
30		dihmeetc.i	. . . . 5 |- I = ( ( DIsoH ` K ) ` W )
31	7, 15, 8, 12, 30	dihmeetbN 36592	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( ( Y ./\ W ) e. B /\ ( Y ./\ W ) .<_ W ) ) -> ( I ` ( X ./\ ( Y ./\ W ) ) ) = ( ( I ` X ) i^i ( I ` ( Y ./\ W ) ) ) )
32	25, 5, 27, 29, 31	syl112anc 1330	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( I ` ( X ./\ ( Y ./\ W ) ) ) = ( ( I ` X ) i^i ( I ` ( Y ./\ W ) ) ) )
33	7, 15	latref 17053	. . . . . 6 |- ( ( K e. Lat /\ W e. B ) -> W .<_ W )
34	4, 14, 33	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> W .<_ W )
35	7, 15, 8, 12, 30	dihmeetbN 36592	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( W e. B /\ W .<_ W ) ) -> ( I ` ( Y ./\ W ) ) = ( ( I ` Y ) i^i ( I ` W ) ) )
36	25, 6, 14, 34, 35	syl112anc 1330	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( I ` ( Y ./\ W ) ) = ( ( I ` Y ) i^i ( I ` W ) ) )
37	36	ineq2d 3814	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( ( I ` X ) i^i ( I ` ( Y ./\ W ) ) ) = ( ( I ` X ) i^i ( ( I ` Y ) i^i ( I ` W ) ) ) )
38	24, 32, 37	3eqtrd 2660	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( ( I ` Y ) i^i ( I ` W ) ) ) )
39		inass 3823	. 2 |- ( ( ( I ` X ) i^i ( I ` Y ) ) i^i ( I ` W ) ) = ( ( I ` X ) i^i ( ( I ` Y ) i^i ( I ` W ) ) )
40	38, 39	syl6eqr 2674	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ ( X ./\ Y ) .<_ W ) -> ( I ` ( X ./\ Y ) ) = ( ( ( I ` X ) i^i ( I ` Y ) ) i^i ( I ` W ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   meetcmee 16945   Latclat 17045   OLcol 34461   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  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-fal 1489  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  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-tendo 36043  df-edring 36045  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518

This theorem is referenced by: (None)