Theorem ltrnset 35404

Description: The set of lattice translations for a fiducial co-atom W. (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
ltrnset.l	|- .<_ = ( le ` K )
ltrnset.j	|- .\/ = ( join ` K )
ltrnset.m	|- ./\ = ( meet ` K )
ltrnset.a	|- A = ( Atoms ` K )
ltrnset.h	|- H = ( LHyp ` K )
ltrnset.d	|- D = ( ( LDil ` K ) ` W )
ltrnset.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
ltrnset	|- ( ( K e. B /\ W e. H ) -> T = { f e. D | A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) } )

Distinct variable groups:   q, p, A   D, f   f, p, q, K   f, W, p, q

Allowed substitution hints:   A( f)   B( f, q, p)   D( q, p)   T( f, q, p)   H( f, q, p)   .\/ ( f, q, p)   .<_ ( f, q, p)   ./\ ( f, q, p)

Proof of Theorem ltrnset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrnset.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
2		ltrnset.l	. . . . 5 |- .<_ = ( le ` K )
3		ltrnset.j	. . . . 5 |- .\/ = ( join ` K )
4		ltrnset.m	. . . . 5 |- ./\ = ( meet ` K )
5		ltrnset.a	. . . . 5 |- A = ( Atoms ` K )
6		ltrnset.h	. . . . 5 |- H = ( LHyp ` K )
7	2, 3, 4, 5, 6	ltrnfset 35403	. . . 4 |- ( K e. B -> ( LTrn ` K ) = ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) )
8	7	fveq1d 6193	. . 3 |- ( K e. B -> ( ( LTrn ` K ) ` W ) = ( ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) ` W ) )
9	1, 8	syl5eq 2668	. 2 |- ( K e. B -> T = ( ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) ` W ) )
10		fveq2 6191	. . . . 5 |- ( w = W -> ( ( LDil ` K ) ` w ) = ( ( LDil ` K ) ` W ) )
11		ltrnset.d	. . . . 5 |- D = ( ( LDil ` K ) ` W )
12	10, 11	syl6eqr 2674	. . . 4 |- ( w = W -> ( ( LDil ` K ) ` w ) = D )
13		breq2 4657	. . . . . . . 8 |- ( w = W -> ( p .<_ w <-> p .<_ W ) )
14	13	notbid 308	. . . . . . 7 |- ( w = W -> ( -. p .<_ w <-> -. p .<_ W ) )
15		breq2 4657	. . . . . . . 8 |- ( w = W -> ( q .<_ w <-> q .<_ W ) )
16	15	notbid 308	. . . . . . 7 |- ( w = W -> ( -. q .<_ w <-> -. q .<_ W ) )
17	14, 16	anbi12d 747	. . . . . 6 |- ( w = W -> ( ( -. p .<_ w /\ -. q .<_ w ) <-> ( -. p .<_ W /\ -. q .<_ W ) ) )
18		oveq2 6658	. . . . . . 7 |- ( w = W -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( p .\/ ( f ` p ) ) ./\ W ) )
19		oveq2 6658	. . . . . . 7 |- ( w = W -> ( ( q .\/ ( f ` q ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ W ) )
20	18, 19	eqeq12d 2637	. . . . . 6 |- ( w = W -> ( ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) <-> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) )
21	17, 20	imbi12d 334	. . . . 5 |- ( w = W -> ( ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) <-> ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) ) )
22	21	2ralbidv 2989	. . . 4 |- ( w = W -> ( A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) <-> A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) ) )
23	12, 22	rabeqbidv 3195	. . 3 |- ( w = W -> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } = { f e. D | A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) } )
24		eqid 2622	. . 3 |- ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) = ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } )
25		fvex 6201	. . . . 5 |- ( ( LDil ` K ) ` W ) e. _V
26	11, 25	eqeltri 2697	. . . 4 |- D e. _V
27	26	rabex 4813	. . 3 |- { f e. D | A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) } e. _V
28	23, 24, 27	fvmpt 6282	. 2 |- ( W e. H -> ( ( w e. H |-> { f e. ( ( LDil ` K ) ` w ) | A. p e. A A. q e. A ( ( -. p .<_ w /\ -. q .<_ w ) -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( q .\/ ( f ` q ) ) ./\ w ) ) } ) ` W ) = { f e. D | A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) } )
29	9, 28	sylan9eq 2676	1 |- ( ( K e. B /\ W e. H ) -> T = { f e. D | A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   LHypclh 35270   LDilcldil 35386   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-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-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-reu 2919  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-ov 6653  df-ltrn 35391

This theorem is referenced by:  isltrn  35405