Theorem cdlemg7fvbwN 35895

Description: Properties of a translation of an element not under W. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 35790? Done with a *ltrn* theorem? (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg4.l	|- .<_ = ( le ` K )
cdlemg4.a	|- A = ( Atoms ` K )
cdlemg4.h	|- H = ( LHyp ` K )
cdlemg4.t	|- T = ( ( LTrn ` K ) ` W )
cdlemg4.b	|- B = ( Base ` K )

Assertion

Ref	Expression
cdlemg7fvbwN	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) -> ( ( F ` X ) e. B /\ -. ( F ` X ) .<_ W ) )

Proof of Theorem cdlemg7fvbwN

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemg4.b	. . . 4 |- B = ( Base ` K )
2		cdlemg4.l	. . . 4 |- .<_ = ( le ` K )
3		eqid 2622	. . . 4 |- ( join ` K ) = ( join ` K )
4		eqid 2622	. . . 4 |- ( meet ` K ) = ( meet ` K )
5		cdlemg4.a	. . . 4 |- A = ( Atoms ` K )
6		cdlemg4.h	. . . 4 |- H = ( LHyp ` K )
7	1, 2, 3, 4, 5, 6	lhpmcvr2 35310	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. r e. A ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) )
8	7	3adant3 1081	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) -> E. r e. A ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) )
9		simp11 1091	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( K e. HL /\ W e. H ) )
10		simp2 1062	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> r e. A )
11		simp3l 1089	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> -. r .<_ W )
12	10, 11	jca 554	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( r e. A /\ -. r .<_ W ) )
13		simp12 1092	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( X e. B /\ -. X .<_ W ) )
14		simp13 1093	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> F e. T )
15		simp3r 1090	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X )
16		cdlemg4.t	. . . . . . 7 |- T = ( ( LTrn ` K ) ` W )
17	6, 16, 2, 3, 5, 4, 1	cdlemg2fv 35887	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( r e. A /\ -. r .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` X ) = ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) )
18	9, 12, 13, 14, 15, 17	syl122anc 1335	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` X ) = ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) )
19		simp11l 1172	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> K e. HL )
20		hllat 34650	. . . . . . 7 |- ( K e. HL -> K e. Lat )
21	19, 20	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> K e. Lat )
22	2, 5, 6, 16	ltrnel 35425	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( r e. A /\ -. r .<_ W ) ) -> ( ( F ` r ) e. A /\ -. ( F ` r ) .<_ W ) )
23	22	simpld 475	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( r e. A /\ -. r .<_ W ) ) -> ( F ` r ) e. A )
24	9, 14, 12, 23	syl3anc 1326	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` r ) e. A )
25	1, 5	atbase 34576	. . . . . . 7 |- ( ( F ` r ) e. A -> ( F ` r ) e. B )
26	24, 25	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` r ) e. B )
27		simp12l 1174	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> X e. B )
28		simp11r 1173	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> W e. H )
29	1, 6	lhpbase 35284	. . . . . . . 8 |- ( W e. H -> W e. B )
30	28, 29	syl 17	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> W e. B )
31	1, 4	latmcl 17052	. . . . . . 7 |- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ( meet ` K ) W ) e. B )
32	21, 27, 30, 31	syl3anc 1326	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( X ( meet ` K ) W ) e. B )
33	1, 3	latjcl 17051	. . . . . 6 |- ( ( K e. Lat /\ ( F ` r ) e. B /\ ( X ( meet ` K ) W ) e. B ) -> ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) e. B )
34	21, 26, 32, 33	syl3anc 1326	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) e. B )
35	18, 34	eqeltrd 2701	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` X ) e. B )
36	22	simprd 479	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( r e. A /\ -. r .<_ W ) ) -> -. ( F ` r ) .<_ W )
37	9, 14, 12, 36	syl3anc 1326	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> -. ( F ` r ) .<_ W )
38	1, 2, 3	latlej1 17060	. . . . . . . 8 |- ( ( K e. Lat /\ ( F ` r ) e. B /\ ( X ( meet ` K ) W ) e. B ) -> ( F ` r ) .<_ ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) )
39	21, 26, 32, 38	syl3anc 1326	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` r ) .<_ ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) )
40	1, 2	lattr 17056	. . . . . . . 8 |- ( ( K e. Lat /\ ( ( F ` r ) e. B /\ ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) e. B /\ W e. B ) ) -> ( ( ( F ` r ) .<_ ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) /\ ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) .<_ W ) -> ( F ` r ) .<_ W ) )
41	21, 26, 34, 30, 40	syl13anc 1328	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( ( ( F ` r ) .<_ ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) /\ ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) .<_ W ) -> ( F ` r ) .<_ W ) )
42	39, 41	mpand 711	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) .<_ W -> ( F ` r ) .<_ W ) )
43	37, 42	mtod 189	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> -. ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) .<_ W )
44	18	breq1d 4663	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( ( F ` X ) .<_ W <-> ( ( F ` r ) ( join ` K ) ( X ( meet ` K ) W ) ) .<_ W ) )
45	43, 44	mtbird 315	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> -. ( F ` X ) .<_ W )
46	35, 45	jca 554	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( ( F ` X ) e. B /\ -. ( F ` X ) .<_ W ) )
47	46	rexlimdv3a 3033	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) -> ( E. r e. A ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) -> ( ( F ` X ) e. B /\ -. ( F ` X ) .<_ W ) ) )
48	8, 47	mpd 15	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ F e. T ) -> ( ( F ` X ) e. B /\ -. ( F ` X ) .<_ W ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387

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

This theorem is referenced by:  cdlemg7fvN  35912