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Theorem idltrn 34103
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 2451 . . 3  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
41, 2, 3idldil 34067 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LDil `  K ) `  W
) )
5 simpll 753 . . . . . 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 760 . . . . . 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 756 . . . . . 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 2451 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
9 eqid 2451 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
10 eqid 2451 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 eqid 2451 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
128, 9, 10, 11, 2lhpmat 33983 . . . . . 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 1217 . . . . 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 33243 . . . . . . . . 9  |-  ( q  e.  ( Atoms `  K
)  ->  q  e.  B )
15 fvresi 6006 . . . . . . . . 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 6209 . . . . . . 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 757 . . . . . . . 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 2451 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
2019, 11hlatjidm 33322 . . . . . . . 8  |-  ( ( K  e.  HL  /\  q  e.  ( Atoms `  K ) )  -> 
( q ( join `  K ) q )  =  q )
2118, 6, 20syl2anc 661 . . . . . . 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 2492 . . . . . 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 6208 . . . . 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 759 . . . . . . . . . 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 33243 . . . . . . . . . 10  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  B )
26 fvresi 6006 . . . . . . . . . 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 6209 . . . . . . . 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 33322 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  -> 
( p ( join `  K ) p )  =  p )
3018, 24, 29syl2anc 661 . . . . . . . 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 2492 . . . . . . 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 6208 . . . . . 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 755 . . . . . . 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 33983 . . . . . . 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 1217 . . . . . 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 2492 . . . . 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 2503 . . . 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 434 . . 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 2907 . 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 34072 . 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 913 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   class class class wbr 4393    _I cid 4732    |` cres 4943   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   meetcmee 15226   0.cp0 15318   Atomscatm 33217   HLchlt 33304   LHypclh 33937   LDilcldil 34053   LTrncltrn 34054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058
This theorem is referenced by:  trlid0  34129  tgrpgrplem  34702  tendoid  34726  tendo0cl  34743  cdlemkid2  34877  cdlemkid3N  34886  cdlemkid4  34887  cdlemkid5  34888  cdlemk35s-id  34891  dva0g  34981  dian0  34993  dia0  35006  dvhgrp  35061  dvh0g  35065  dvheveccl  35066  dvhopN  35070  dihmeetlem4preN  35260
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