Theorem ltrnldil 35408

Description: A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)

Hypotheses

Ref	Expression
ltrnldil.h	|- H = ( LHyp ` K )
ltrnldil.d	|- D = ( ( LDil ` K ) ` W )
ltrnldil.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
ltrnldil	|- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. D )

Proof of Theorem ltrnldil

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
2		eqid 2622	. . 3 |- ( join ` K ) = ( join ` K )
3		eqid 2622	. . 3 |- ( meet ` K ) = ( meet ` K )
4		eqid 2622	. . 3 |- ( Atoms ` K ) = ( Atoms ` K )
5		ltrnldil.h	. . 3 |- H = ( LHyp ` K )
6		ltrnldil.d	. . 3 |- D = ( ( LDil ` K ) ` W )
7		ltrnldil.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
8	1, 2, 3, 4, 5, 6, 7	isltrn 35405	. 2 |- ( ( K e. V /\ W e. H ) -> ( F e. T <-> ( F e. D /\ A. p e. ( Atoms ` K ) A. q e. ( Atoms ` K ) ( ( -. p ( le ` K ) W /\ -. q ( le ` K ) W ) -> ( ( p ( join ` K ) ( F ` p ) ) ( meet ` K ) W ) = ( ( q ( join ` K ) ( F ` q ) ) ( meet ` K ) W ) ) ) ) )
9	8	simprbda 653	1 |- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. D )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   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:  ltrnlaut  35409  ltrnval1  35420  ltrncnv  35432  ltrnco  36007