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Theorem idldil 30229
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 2388 . . . 4  |-  ( LAut `  K )  =  (
LAut `  K )
31, 2idlaut 30211 . . 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 5864 . . . . 5  |-  ( x  e.  B  ->  (
(  _I  |`  B ) `
 x )  =  x )
65a1d 23 . . . 4  |-  ( x  e.  B  ->  (
x ( le `  K ) W  -> 
( (  _I  |`  B ) `
 x )  =  x ) )
76rgen 2715 . . 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 2388 . . 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 30225 . 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 1649    e. wcel 1717   A.wral 2650   class class class wbr 4154    _I cid 4435    |` cres 4821   ` cfv 5395   Basecbs 13397   lecple 13464   LHypclh 30099   LAutclaut 30100   LDilcldil 30215
This theorem is referenced by:  idltrn  30265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-map 6957  df-laut 30104  df-ldil 30219
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