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Theorem idltrn 29618
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 2284 . . 3  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
41, 2, 3idldil 29582 . 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 2284 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
9 eqid 2284 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
10 eqid 2284 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 eqid 2284 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
128, 9, 10, 11, 2lhpmat 29498 . . . . . 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 28758 . . . . . . . . 9  |-  ( q  e.  ( Atoms `  K
)  ->  q  e.  B )
15 fvresi 5673 . . . . . . . . 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 5836 . . . . . . 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 2284 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
2019, 11hlatjidm 28837 . . . . . . . 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 2316 . . . . . 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 5835 . . . . 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 28758 . . . . . . . . . 10  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
26 fvresi 5673 . . . . . . . . . 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 5836 . . . . . . . 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 28837 . . . . . . . . 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 2316 . . . . . . 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 5835 . . . . . 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 29498 . . . . . . 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 2316 . . . . 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 2327 . . . 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 2636 . 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 29587 . 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 1623    e. wcel 1685   A.wral 2544   class class class wbr 4024    _I cid 4303    |` cres 4690   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   0.cp0 14139   Atomscatm 28732   HLchlt 28819   LHypclh 29452   LDilcldil 29568   LTrncltrn 29569
This theorem is referenced by:  trlid0  29644  tgrpgrplem  30217  tendoid  30241  tendo0cl  30258  cdlemkid2  30392  cdlemkid3N  30401  cdlemkid4  30402  cdlemkid5  30403  cdlemk35s-id  30406  dva0g  30496  dian0  30508  dia0  30521  dvhgrp  30576  dvh0g  30580  dvheveccl  30581  dvhopN  30585  dihmeetlem4preN  30775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-lat 14148  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-lhyp 29456  df-laut 29457  df-ldil 29572  df-ltrn 29573
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