Theorem idldil 35400

Description: The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)

Hypotheses

Ref	Expression
idldil.b	|- B = ( Base ` K )
idldil.h	|- H = ( LHyp ` K )
idldil.d	|- D = ( ( LDil ` K ) ` W )

Assertion

Ref	Expression
idldil	|- ( ( K e. A /\ W e. H ) -> ( _I |` B ) e. D )

Proof of Theorem idldil

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idldil.b	. . . 4 |- B = ( Base ` K )
2		eqid 2622	. . . 4 |- ( LAut ` K ) = ( LAut ` K )
3	1, 2	idlaut 35382	. . 3 |- ( K e. A -> ( _I |` B ) e. ( LAut ` K ) )
4	3	adantr 481	. 2 |- ( ( K e. A /\ W e. H ) -> ( _I |` B ) e. ( LAut ` K ) )
5		fvresi 6439	. . . . 5 |- ( x e. B -> ( ( _I |` B ) ` x ) = x )
6	5	a1d 25	. . . 4 |- ( x e. B -> ( x ( le ` K ) W -> ( ( _I |` B ) ` x ) = x ) )
7	6	rgen 2922	. . 3 |- A. x e. B ( x ( le ` K ) W -> ( ( _I |` B ) ` x ) = x )
8	7	a1i 11	. 2 |- ( ( K e. A /\ W e. H ) -> A. x e. B ( x ( le ` K ) W -> ( ( _I |` B ) ` x ) = x ) )
9		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
10		idldil.h	. . 3 |- H = ( LHyp ` K )
11		idldil.d	. . 3 |- D = ( ( LDil ` K ) ` W )
12	1, 9, 10, 2, 11	isldil 35396	. 2 |- ( ( K e. A /\ W e. H ) -> ( ( _I |` B ) e. D <-> ( ( _I |` B ) e. ( LAut ` K ) /\ A. x e. B ( x ( le ` K ) W -> ( ( _I |` B ) ` x ) = x ) ) ) )
13	4, 8, 12	mpbir2and 957	1 |- ( ( K e. A /\ W e. H ) -> ( _I |` B ) e. D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   _I cid 5023   |` cres 5116   ` 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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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-oprab 6654  df-mpt2 6655  df-map 7859  df-laut 35275  df-ldil 35390

This theorem is referenced by:  idltrn  35436