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Theorem idltrn 30636
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 2408 . . 3  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
41, 2, 3idldil 30600 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LDil `  K ) `  W
) )
5 simpll 731 . . . . . 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 738 . . . . . 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 734 . . . . . 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 2408 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
9 eqid 2408 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
10 eqid 2408 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 eqid 2408 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
128, 9, 10, 11, 2lhpmat 30516 . . . . . 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 1182 . . . . 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 29776 . . . . . . . . 9  |-  ( q  e.  ( Atoms `  K
)  ->  q  e.  B )
15 fvresi 5887 . . . . . . . . 9  |-  ( q  e.  B  ->  (
(  _I  |`  B ) `
 q )  =  q )
166, 14, 153syl 19 . . . . . . . 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 6060 . . . . . . 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 735 . . . . . . . 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 2408 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
2019, 11hlatjidm 29855 . . . . . . . 8  |-  ( ( K  e.  HL  /\  q  e.  ( Atoms `  K ) )  -> 
( q ( join `  K ) q )  =  q )
2118, 6, 20syl2anc 643 . . . . . . 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 2440 . . . . . 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 6059 . . . . 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 737 . . . . . . . . . 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 29776 . . . . . . . . . 10  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
26 fvresi 5887 . . . . . . . . . 10  |-  ( p  e.  B  ->  (
(  _I  |`  B ) `
 p )  =  p )
2724, 25, 263syl 19 . . . . . . . . 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 6060 . . . . . . . 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 29855 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  -> 
( p ( join `  K ) p )  =  p )
3018, 24, 29syl2anc 643 . . . . . . . 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 2440 . . . . . . 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 6059 . . . . . 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 733 . . . . . . 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 30516 . . . . . . 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 1182 . . . . . 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 2440 . . . . 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 2451 . . . 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 424 . . 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 2762 . 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 30605 . 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 889 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   class class class wbr 4176    _I cid 4457    |` cres 4843   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   0.cp0 14425   Atomscatm 29750   HLchlt 29837   LHypclh 30470   LDilcldil 30586   LTrncltrn 30587
This theorem is referenced by:  trlid0  30662  tgrpgrplem  31235  tendoid  31259  tendo0cl  31276  cdlemkid2  31410  cdlemkid3N  31419  cdlemkid4  31420  cdlemkid5  31421  cdlemk35s-id  31424  dva0g  31514  dian0  31526  dia0  31539  dvhgrp  31594  dvh0g  31598  dvheveccl  31599  dvhopN  31603  dihmeetlem4preN  31793
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-lat 14434  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591
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