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Theorem idldil 30838
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.
StepHypRef Expression
1 idldil.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2435 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
31, 2idlaut 30820 . . 3  |-  ( K  e.  A  ->  (  _I  |`  B )  e.  ( LAut `  K
) )
43adantr 452 . 2  |-  ( ( K  e.  A  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( LAut `  K
) )
5 fvresi 5916 . . . . 5  |-  ( x  e.  B  ->  (
(  _I  |`  B ) `
 x )  =  x )
65a1d 23 . . . 4  |-  ( x  e.  B  ->  (
x ( le `  K ) W  -> 
( (  _I  |`  B ) `
 x )  =  x ) )
76rgen 2763 . . 3  |-  A. x  e.  B  ( x
( le `  K
) W  ->  (
(  _I  |`  B ) `
 x )  =  x )
87a1i 11 . 2  |-  ( ( K  e.  A  /\  W  e.  H )  ->  A. x  e.  B  ( x ( le
`  K ) W  ->  ( (  _I  |`  B ) `  x
)  =  x ) )
9 eqid 2435 . . 3  |-  ( le
`  K )  =  ( le `  K
)
10 idldil.h . . 3  |-  H  =  ( LHyp `  K
)
11 idldil.d . . 3  |-  D  =  ( ( LDil `  K
) `  W )
121, 9, 10, 2, 11isldil 30834 . 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 ) ) ) )
134, 8, 12mpbir2and 889 1  |-  ( ( K  e.  A  /\  W  e.  H )  ->  (  _I  |`  B )  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   LHypclh 30708   LAutclaut 30709   LDilcldil 30824
This theorem is referenced by:  idltrn  30874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-laut 30713  df-ldil 30828
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