Theorem cdlemftr3 35853

Description: Special case of cdlemf 35851 showing existence of non-identity translation with trace different from any 3 given lattice elements. (Contributed by NM, 24-Jul-2013.)

Hypotheses

Ref	Expression
cdlemftr.b	|- B = ( Base ` K )
cdlemftr.h	|- H = ( LHyp ` K )
cdlemftr.t	|- T = ( ( LTrn ` K ) ` W )
cdlemftr.r	|- R = ( ( trL ` K ) ` W )

Assertion

Ref	Expression
cdlemftr3	|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) )

Distinct variable groups:   f, X   f, Y   f, Z   f, H   f, K   R, f   T, f   f, W

Allowed substitution hint:   B( f)

Proof of Theorem cdlemftr3

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( le ` K ) = ( le ` K )
2		eqid 2622	. . . . 5 |- ( Atoms ` K ) = ( Atoms ` K )
3		cdlemftr.h	. . . . 5 |- H = ( LHyp ` K )
4	1, 2, 3	lhpexle3 35298	. . . 4 |- ( ( K e. HL /\ W e. H ) -> E. u e. ( Atoms ` K ) ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) )
5		df-rex 2918	. . . 4 |- ( E. u e. ( Atoms ` K ) ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) )
6	4, 5	sylib 208	. . 3 |- ( ( K e. HL /\ W e. H ) -> E. u ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) )
7		cdlemftr.b	. . . . . . . . 9 |- B = ( Base ` K )
8		cdlemftr.t	. . . . . . . . 9 |- T = ( ( LTrn ` K ) ` W )
9		cdlemftr.r	. . . . . . . . 9 |- R = ( ( trL ` K ) ` W )
10	7, 1, 2, 3, 8, 9	cdlemfnid 35852	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ u ( le ` K ) W ) ) -> E. f e. T ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) )
11	10	adantrrr 761	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> E. f e. T ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) )
12		eqcom 2629	. . . . . . . . 9 |- ( ( R ` f ) = u <-> u = ( R ` f ) )
13	12	anbi1i 731	. . . . . . . 8 |- ( ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) <-> ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) )
14	13	rexbii 3041	. . . . . . 7 |- ( E. f e. T ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) <-> E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) )
15	11, 14	sylib 208	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) )
16		simprrr 805	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> ( u =/= X /\ u =/= Y /\ u =/= Z ) )
17	15, 16	jca 554	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) )
18	17	ex 450	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) -> ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) )
19	18	eximdv 1846	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( E. u ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) -> E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) )
20	6, 19	mpd 15	. 2 |- ( ( K e. HL /\ W e. H ) -> E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) )
21		rexcom4 3225	. . 3 |- ( E. f e. T E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u E. f e. T ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) )
22		anass 681	. . . . . 6 |- ( ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( u = ( R ` f ) /\ ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) )
23	22	exbii 1774	. . . . 5 |- ( E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u ( u = ( R ` f ) /\ ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) )
24		fvex 6201	. . . . . 6 |- ( R ` f ) e. _V
25		neeq1 2856	. . . . . . . 8 |- ( u = ( R ` f ) -> ( u =/= X <-> ( R ` f ) =/= X ) )
26		neeq1 2856	. . . . . . . 8 |- ( u = ( R ` f ) -> ( u =/= Y <-> ( R ` f ) =/= Y ) )
27		neeq1 2856	. . . . . . . 8 |- ( u = ( R ` f ) -> ( u =/= Z <-> ( R ` f ) =/= Z ) )
28	25, 26, 27	3anbi123d 1399	. . . . . . 7 |- ( u = ( R ` f ) -> ( ( u =/= X /\ u =/= Y /\ u =/= Z ) <-> ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) )
29	28	anbi2d 740	. . . . . 6 |- ( u = ( R ` f ) -> ( ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) )
30	24, 29	ceqsexv 3242	. . . . 5 |- ( E. u ( u = ( R ` f ) /\ ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) <-> ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) )
31	23, 30	bitri 264	. . . 4 |- ( E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) )
32	31	rexbii 3041	. . 3 |- ( E. f e. T E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) )
33		r19.41v 3089	. . . 4 |- ( E. f e. T ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) )
34	33	exbii 1774	. . 3 |- ( E. u E. f e. T ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) )
35	21, 32, 34	3bitr3ri 291	. 2 |- ( E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) )
36	20, 35	sylib 208	1 |- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

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:  cdlemftr2  35854  cdlemk26-3  36194  cdlemk11t  36234