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Theorem idltrn 30412
Description: The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
idltrn.b  |-  B  =  ( Base `  K
)
idltrn.h  |-  H  =  ( LHyp `  K
)
idltrn.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
idltrn  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )

Proof of Theorem idltrn
Dummy variables  q  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idltrn.b . . 3  |-  B  =  ( Base `  K
)
2 idltrn.h . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2285 . . 3  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
41, 2, 3idldil 30376 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LDil `  K ) `  W
) )
5 simpll 730 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simplrr 737 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  q  e.  ( Atoms `  K )
)
7 simprr 733 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  -.  q
( le `  K
) W )
8 eqid 2285 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
9 eqid 2285 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
10 eqid 2285 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 eqid 2285 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
128, 9, 10, 11, 2lhpmat 30292 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( q  e.  ( Atoms `  K )  /\  -.  q ( le
`  K ) W ) )  ->  (
q ( meet `  K
) W )  =  ( 0. `  K
) )
135, 6, 7, 12syl12anc 1180 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( q
( meet `  K ) W )  =  ( 0. `  K ) )
141, 11atbase 29552 . . . . . . . . 9  |-  ( q  e.  ( Atoms `  K
)  ->  q  e.  B )
15 fvresi 5713 . . . . . . . . 9  |-  ( q  e.  B  ->  (
(  _I  |`  B ) `
 q )  =  q )
166, 14, 153syl 18 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( (  _I  |`  B ) `  q )  =  q )
1716oveq2d 5876 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( q
( join `  K )
( (  _I  |`  B ) `
 q ) )  =  ( q (
join `  K )
q ) )
18 simplll 734 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  K  e.  HL )
19 eqid 2285 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
2019, 11hlatjidm 29631 . . . . . . . 8  |-  ( ( K  e.  HL  /\  q  e.  ( Atoms `  K ) )  -> 
( q ( join `  K ) q )  =  q )
2118, 6, 20syl2anc 642 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( q
( join `  K )
q )  =  q )
2217, 21eqtrd 2317 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( q
( join `  K )
( (  _I  |`  B ) `
 q ) )  =  q )
2322oveq1d 5875 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( (
q ( join `  K
) ( (  _I  |`  B ) `  q
) ) ( meet `  K ) W )  =  ( q (
meet `  K ) W ) )
24 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  p  e.  ( Atoms `  K )
)
251, 11atbase 29552 . . . . . . . . . 10  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
26 fvresi 5713 . . . . . . . . . 10  |-  ( p  e.  B  ->  (
(  _I  |`  B ) `
 p )  =  p )
2724, 25, 263syl 18 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( (  _I  |`  B ) `  p )  =  p )
2827oveq2d 5876 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( p
( join `  K )
( (  _I  |`  B ) `
 p ) )  =  ( p (
join `  K )
p ) )
2919, 11hlatjidm 29631 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  -> 
( p ( join `  K ) p )  =  p )
3018, 24, 29syl2anc 642 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( p
( join `  K )
p )  =  p )
3128, 30eqtrd 2317 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( p
( join `  K )
( (  _I  |`  B ) `
 p ) )  =  p )
3231oveq1d 5875 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( (
p ( join `  K
) ( (  _I  |`  B ) `  p
) ) ( meet `  K ) W )  =  ( p (
meet `  K ) W ) )
33 simprl 732 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  -.  p
( le `  K
) W )
348, 9, 10, 11, 2lhpmat 30292 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (
p ( meet `  K
) W )  =  ( 0. `  K
) )
355, 24, 33, 34syl12anc 1180 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( p
( meet `  K ) W )  =  ( 0. `  K ) )
3632, 35eqtrd 2317 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( (
p ( join `  K
) ( (  _I  |`  B ) `  p
) ) ( meet `  K ) W )  =  ( 0. `  K ) )
3713, 23, 363eqtr4rd 2328 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K )
) )  /\  ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W ) )  ->  ( (
p ( join `  K
) ( (  _I  |`  B ) `  p
) ) ( meet `  K ) W )  =  ( ( q ( join `  K
) ( (  _I  |`  B ) `  q
) ) ( meet `  K ) W ) )
3837ex 423 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  ( ( -.  p ( le `  K ) W  /\  -.  q ( le `  K ) W )  ->  ( ( p ( join `  K
) ( (  _I  |`  B ) `  p
) ) ( meet `  K ) W )  =  ( ( q ( join `  K
) ( (  _I  |`  B ) `  q
) ) ( meet `  K ) W ) ) )
3938ralrimivva 2637 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  A. p  e.  (
Atoms `  K ) A. q  e.  ( Atoms `  K ) ( ( -.  p ( le
`  K ) W  /\  -.  q ( le `  K ) W )  ->  (
( p ( join `  K ) ( (  _I  |`  B ) `  p ) ) (
meet `  K ) W )  =  ( ( q ( join `  K ) ( (  _I  |`  B ) `  q ) ) (
meet `  K ) W ) ) )
40 idltrn.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
418, 19, 9, 11, 2, 3, 40isltrn 30381 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( (  _I  |`  B )  e.  T  <->  ( (  _I  |`  B )  e.  ( ( LDil `  K
) `  W )  /\  A. p  e.  (
Atoms `  K ) A. q  e.  ( Atoms `  K ) ( ( -.  p ( le
`  K ) W  /\  -.  q ( le `  K ) W )  ->  (
( p ( join `  K ) ( (  _I  |`  B ) `  p ) ) (
meet `  K ) W )  =  ( ( q ( join `  K ) ( (  _I  |`  B ) `  q ) ) (
meet `  K ) W ) ) ) ) )
424, 39, 41mpbir2and 888 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545   class class class wbr 4025    _I cid 4306    |` cres 4693   ` cfv 5257  (class class class)co 5860   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   0.cp0 14145   Atomscatm 29526   HLchlt 29613   LHypclh 30246   LDilcldil 30362   LTrncltrn 30363
This theorem is referenced by:  trlid0  30438  tgrpgrplem  31011  tendoid  31035  tendo0cl  31052  cdlemkid2  31186  cdlemkid3N  31195  cdlemkid4  31196  cdlemkid5  31197  cdlemk35s-id  31200  dva0g  31290  dian0  31302  dia0  31315  dvhgrp  31370  dvh0g  31374  dvheveccl  31375  dvhopN  31379  dihmeetlem4preN  31569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-map 6776  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-lat 14154  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367
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