Theorem isltrn 35405

Description: The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (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
isltrn	|- ( ( K e. B /\ W e. H ) -> ( F e. 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   K, p, q   W, p, q   F, p, q

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

Proof of Theorem isltrn

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrnset.l	. . . 4 |- .<_ = ( le ` K )
2		ltrnset.j	. . . 4 |- .\/ = ( join ` K )
3		ltrnset.m	. . . 4 |- ./\ = ( meet ` K )
4		ltrnset.a	. . . 4 |- A = ( Atoms ` K )
5		ltrnset.h	. . . 4 |- H = ( LHyp ` K )
6		ltrnset.d	. . . 4 |- D = ( ( LDil ` K ) ` W )
7		ltrnset.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
8	1, 2, 3, 4, 5, 6, 7	ltrnset 35404	. . 3 |- ( ( 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 ) ) } )
9	8	eleq2d 2687	. 2 |- ( ( K e. B /\ W e. H ) -> ( F e. T <-> F e. { f e. D | A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) } ) )
10		fveq1 6190	. . . . . . . 8 |- ( f = F -> ( f ` p ) = ( F ` p ) )
11	10	oveq2d 6666	. . . . . . 7 |- ( f = F -> ( p .\/ ( f ` p ) ) = ( p .\/ ( F ` p ) ) )
12	11	oveq1d 6665	. . . . . 6 |- ( f = F -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( p .\/ ( F ` p ) ) ./\ W ) )
13		fveq1 6190	. . . . . . . 8 |- ( f = F -> ( f ` q ) = ( F ` q ) )
14	13	oveq2d 6666	. . . . . . 7 |- ( f = F -> ( q .\/ ( f ` q ) ) = ( q .\/ ( F ` q ) ) )
15	14	oveq1d 6665	. . . . . 6 |- ( f = F -> ( ( q .\/ ( f ` q ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) )
16	12, 15	eqeq12d 2637	. . . . 5 |- ( f = F -> ( ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) <-> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) )
17	16	imbi2d 330	. . . 4 |- ( f = F -> ( ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( f ` p ) ) ./\ W ) = ( ( q .\/ ( f ` q ) ) ./\ W ) ) <-> ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) )
18	17	2ralbidv 2989	. . 3 |- ( f = F -> ( 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 ) ) ) )
19	18	elrab 3363	. 2 |- ( F e. { f e. D | 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 ) ) ) )
20	9, 19	syl6bb 276	1 |- ( ( K e. B /\ W e. H ) -> ( F e. 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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   ` 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:  isltrn2N  35406  ltrnu  35407  ltrnldil  35408  ltrncnv  35432  idltrn  35436  cdleme50ltrn  35845  ltrnco  36007