Theorem cdlemf 35851

Description: Lemma F in [Crawley] p. 116. If u is an atom under w, there exists a translation whose trace is u. (Contributed by NM, 12-Apr-2013.)

Hypotheses

Ref	Expression
cdlemf.l	|- .<_ = ( le ` K )
cdlemf.a	|- A = ( Atoms ` K )
cdlemf.h	|- H = ( LHyp ` K )
cdlemf.t	|- T = ( ( LTrn ` K ) ` W )
cdlemf.r	|- R = ( ( trL ` K ) ` W )

Assertion

Ref	Expression
cdlemf	|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> E. f e. T ( R ` f ) = U )

Distinct variable groups:   A, f   f, H   f, K   .<_ , f   T, f   U, f   f, W

Allowed substitution hint:   R( f)

Proof of Theorem cdlemf

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemf.l	. . 3 |- .<_ = ( le ` K )
2		eqid 2622	. . 3 |- ( join ` K ) = ( join ` K )
3		cdlemf.a	. . 3 |- A = ( Atoms ` K )
4		cdlemf.h	. . 3 |- H = ( LHyp ` K )
5		eqid 2622	. . 3 |- ( meet ` K ) = ( meet ` K )
6	1, 2, 3, 4, 5	cdlemf2 35850	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> E. p e. A E. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) )
7		simp1l 1085	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> ( K e. HL /\ W e. H ) )
8		simp2l 1087	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> p e. A )
9		simp3ll 1132	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> -. p .<_ W )
10		simp2r 1088	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> q e. A )
11		simp3lr 1133	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> -. q .<_ W )
12		cdlemf.t	. . . . . . 7 |- T = ( ( LTrn ` K ) ` W )
13	1, 3, 4, 12	cdleme50ex 35847	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p .<_ W ) /\ ( q e. A /\ -. q .<_ W ) ) -> E. f e. T ( f ` p ) = q )
14	7, 8, 9, 10, 11, 13	syl122anc 1335	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> E. f e. T ( f ` p ) = q )
15		simp3r 1090	. . . . . . . . . . . . 13 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> ( f ` p ) = q )
16	15	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> ( p ( join ` K ) ( f ` p ) ) = ( p ( join ` K ) q ) )
17	16	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> ( ( p ( join ` K ) ( f ` p ) ) ( meet ` K ) W ) = ( ( p ( join ` K ) q ) ( meet ` K ) W ) )
18		simp11 1091	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> ( K e. HL /\ W e. H ) )
19		simp3l 1089	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> f e. T )
20		simp13l 1176	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> p e. A )
21		simp2ll 1128	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> -. p .<_ W )
22		cdlemf.r	. . . . . . . . . . . . 13 |- R = ( ( trL ` K ) ` W )
23	1, 2, 5, 3, 4, 12, 22	trlval2 35450	. . . . . . . . . . . 12 |- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ ( p e. A /\ -. p .<_ W ) ) -> ( R ` f ) = ( ( p ( join ` K ) ( f ` p ) ) ( meet ` K ) W ) )
24	18, 19, 20, 21, 23	syl112anc 1330	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> ( R ` f ) = ( ( p ( join ` K ) ( f ` p ) ) ( meet ` K ) W ) )
25		simp2r 1088	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) )
26	17, 24, 25	3eqtr4d 2666	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) /\ ( f e. T /\ ( f ` p ) = q ) ) -> ( R ` f ) = U )
27	26	3exp 1264	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( p e. A /\ q e. A ) ) -> ( ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) -> ( ( f e. T /\ ( f ` p ) = q ) -> ( R ` f ) = U ) ) )
28	27	3expia 1267	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> ( ( p e. A /\ q e. A ) -> ( ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) -> ( ( f e. T /\ ( f ` p ) = q ) -> ( R ` f ) = U ) ) ) )
29	28	3imp 1256	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> ( ( f e. T /\ ( f ` p ) = q ) -> ( R ` f ) = U ) )
30	29	expd 452	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> ( f e. T -> ( ( f ` p ) = q -> ( R ` f ) = U ) ) )
31	30	reximdvai 3015	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> ( E. f e. T ( f ` p ) = q -> E. f e. T ( R ` f ) = U ) )
32	14, 31	mpd 15	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ ( p e. A /\ q e. A ) /\ ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) ) -> E. f e. T ( R ` f ) = U )
33	32	3exp 1264	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> ( ( p e. A /\ q e. A ) -> ( ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) -> E. f e. T ( R ` f ) = U ) ) )
34	33	rexlimdvv 3037	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> ( E. p e. A E. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) /\ U = ( ( p ( join ` K ) q ) ( meet ` K ) W ) ) -> E. f e. T ( R ` f ) = U ) )
35	6, 34	mpd 15	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> E. f e. T ( R ` f ) = U )

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   lecple 15948   joincjn 16944   meetcmee 16945   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:  cdlemfnid  35852  trlord  35857  dih1dimb2  36530