Theorem ldilset 35395

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

Hypotheses

Ref	Expression
ldilset.b	|- B = ( Base ` K )
ldilset.l	|- .<_ = ( le ` K )
ldilset.h	|- H = ( LHyp ` K )
ldilset.i	|- I = ( LAut ` K )
ldilset.d	|- D = ( ( LDil ` K ) ` W )

Assertion

Ref	Expression
ldilset	|- ( ( K e. C /\ W e. H ) -> D = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )

Distinct variable groups:   x, B   f, I   x, f, K   f, W, x

Allowed substitution hints:   B( f)   C( x, f)   D( x, f)   H( x, f)   I( x)   .<_ ( x, f)

Proof of Theorem ldilset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldilset.d	. 2 |- D = ( ( LDil ` K ) ` W )
2		ldilset.b	. . . . 5 |- B = ( Base ` K )
3		ldilset.l	. . . . 5 |- .<_ = ( le ` K )
4		ldilset.h	. . . . 5 |- H = ( LHyp ` K )
5		ldilset.i	. . . . 5 |- I = ( LAut ` K )
6	2, 3, 4, 5	ldilfset 35394	. . . 4 |- ( K e. C -> ( LDil ` K ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) )
7	6	fveq1d 6193	. . 3 |- ( K e. C -> ( ( LDil ` K ) ` W ) = ( ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) ` W ) )
8		breq2 4657	. . . . . . 7 |- ( w = W -> ( x .<_ w <-> x .<_ W ) )
9	8	imbi1d 331	. . . . . 6 |- ( w = W -> ( ( x .<_ w -> ( f ` x ) = x ) <-> ( x .<_ W -> ( f ` x ) = x ) ) )
10	9	ralbidv 2986	. . . . 5 |- ( w = W -> ( A. x e. B ( x .<_ w -> ( f ` x ) = x ) <-> A. x e. B ( x .<_ W -> ( f ` x ) = x ) ) )
11	10	rabbidv 3189	. . . 4 |- ( w = W -> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )
12		eqid 2622	. . . 4 |- ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } )
13		fvex 6201	. . . . . 6 |- ( LAut ` K ) e. _V
14	5, 13	eqeltri 2697	. . . . 5 |- I e. _V
15	14	rabex 4813	. . . 4 |- { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } e. _V
16	11, 12, 15	fvmpt 6282	. . 3 |- ( W e. H -> ( ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) ` W ) = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )
17	7, 16	sylan9eq 2676	. 2 |- ( ( K e. C /\ W e. H ) -> ( ( LDil ` K ) ` W ) = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )
18	1, 17	syl5eq 2668	1 |- ( ( K e. C /\ W e. H ) -> D = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )

Syntax hints:   -> 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   Basecbs 15857   lecple 15948   LHypclh 35270   LAutclaut 35271   LDilcldil 35386

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-ldil 35390

This theorem is referenced by:  isldil  35396