Theorem ldilval 35399

Description: Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)

Hypotheses

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

Assertion

Ref	Expression
ldilval	|- ( ( ( K e. V /\ W e. H ) /\ F e. D /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )

Proof of Theorem ldilval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldilval.b	. . . . 5 |- B = ( Base ` K )
2		ldilval.l	. . . . 5 |- .<_ = ( le ` K )
3		ldilval.h	. . . . 5 |- H = ( LHyp ` K )
4		eqid 2622	. . . . 5 |- ( LAut ` K ) = ( LAut ` K )
5		ldilval.d	. . . . 5 |- D = ( ( LDil ` K ) ` W )
6	1, 2, 3, 4, 5	isldil 35396	. . . 4 |- ( ( K e. V /\ W e. H ) -> ( F e. D <-> ( F e. ( LAut ` K ) /\ A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) ) )
7		simpr 477	. . . 4 |- ( ( F e. ( LAut ` K ) /\ A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) -> A. x e. B ( x .<_ W -> ( F ` x ) = x ) )
8	6, 7	syl6bi 243	. . 3 |- ( ( K e. V /\ W e. H ) -> ( F e. D -> A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) )
9		breq1 4656	. . . . . 6 |- ( x = X -> ( x .<_ W <-> X .<_ W ) )
10		fveq2 6191	. . . . . . 7 |- ( x = X -> ( F ` x ) = ( F ` X ) )
11		id 22	. . . . . . 7 |- ( x = X -> x = X )
12	10, 11	eqeq12d 2637	. . . . . 6 |- ( x = X -> ( ( F ` x ) = x <-> ( F ` X ) = X ) )
13	9, 12	imbi12d 334	. . . . 5 |- ( x = X -> ( ( x .<_ W -> ( F ` x ) = x ) <-> ( X .<_ W -> ( F ` X ) = X ) ) )
14	13	rspccv 3306	. . . 4 |- ( A. x e. B ( x .<_ W -> ( F ` x ) = x ) -> ( X e. B -> ( X .<_ W -> ( F ` X ) = X ) ) )
15	14	impd 447	. . 3 |- ( A. x e. B ( x .<_ W -> ( F ` x ) = x ) -> ( ( X e. B /\ X .<_ W ) -> ( F ` X ) = X ) )
16	8, 15	syl6 35	. 2 |- ( ( K e. V /\ W e. H ) -> ( F e. D -> ( ( X e. B /\ X .<_ W ) -> ( F ` X ) = X ) ) )
17	16	3imp 1256	1 |- ( ( ( K e. V /\ W e. H ) /\ F e. D /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` 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:  ldilcnv  35401  ldilco  35402  ltrnval1  35420