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Theorem idltrn 29028
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
StepHypRef Expression
1 idltrn.b . . 3  |-  B  =  ( Base `  K
)
2 idltrn.h . . 3  |-  H  =  ( LHyp `  K
)
3 eqid 2253 . . 3  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
41, 2, 3idldil 28992 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LDil `  K ) `  W
) )
5 simpll 733 . . . . . 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 740 . . . . . 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 736 . . . . . 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 2253 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
9 eqid 2253 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
10 eqid 2253 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 eqid 2253 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
128, 9, 10, 11, 2lhpmat 28908 . . . . . 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 1185 . . . . 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 28168 . . . . . . . . 9  |-  ( q  e.  ( Atoms `  K
)  ->  q  e.  B )
15 fvresi 5563 . . . . . . . . 9  |-  ( q  e.  B  ->  (
(  _I  |`  B ) `
 q )  =  q )
166, 14, 153syl 20 . . . . . . . 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 5726 . . . . . . 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 737 . . . . . . . 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 2253 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
2019, 11hlatjidm 28247 . . . . . . . 8  |-  ( ( K  e.  HL  /\  q  e.  ( Atoms `  K ) )  -> 
( q ( join `  K ) q )  =  q )
2118, 6, 20syl2anc 645 . . . . . . 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 2285 . . . . . 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 5725 . . . . 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 739 . . . . . . . . . 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 28168 . . . . . . . . . 10  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
26 fvresi 5563 . . . . . . . . . 10  |-  ( p  e.  B  ->  (
(  _I  |`  B ) `
 p )  =  p )
2724, 25, 263syl 20 . . . . . . . . 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 5726 . . . . . . . 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 28247 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  -> 
( p ( join `  K ) p )  =  p )
3018, 24, 29syl2anc 645 . . . . . . . 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 2285 . . . . . . 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 5725 . . . . . 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 735 . . . . . . 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 28908 . . . . . . 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 1185 . . . . . 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 2285 . . . . 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 2296 . . . 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 425 . . 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 2597 . 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 28997 . 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 893 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509   class class class wbr 3920    _I cid 4197    |` cres 4582   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   0.cp0 13987   Atomscatm 28142   HLchlt 28229   LHypclh 28862   LDilcldil 28978   LTrncltrn 28979
This theorem is referenced by:  trlid0  29054  tgrpgrplem  29627  tendoid  29651  tendo0cl  29668  cdlemkid2  29802  cdlemkid3N  29811  cdlemkid4  29812  cdlemkid5  29813  cdlemk35s-id  29816  dva0g  29906  dian0  29918  dia0  29931  dvhgrp  29986  dvh0g  29990  dvheveccl  29991  dvhopN  29995  dihmeetlem4preN  30185
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-lat 13996  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983
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