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Theorem dihmeetlem11N 34802
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem9.b  |-  B  =  ( Base `  K
)
dihmeetlem9.l  |-  .<_  =  ( le `  K )
dihmeetlem9.h  |-  H  =  ( LHyp `  K
)
dihmeetlem9.j  |-  .\/  =  ( join `  K )
dihmeetlem9.m  |-  ./\  =  ( meet `  K )
dihmeetlem9.a  |-  A  =  ( Atoms `  K )
dihmeetlem9.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihmeetlem9.s  |-  .(+)  =  (
LSSum `  U )
dihmeetlem9.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihmeetlem11N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( (
I `  ( ( X  ./\  Y )  .\/  p ) )  i^i  ( I `  Y
) )  =  ( ( I `  X
)  i^i  ( I `  Y ) ) )

Proof of Theorem dihmeetlem11N
StepHypRef Expression
1 dihmeetlem9.b . . . 4  |-  B  =  ( Base `  K
)
2 dihmeetlem9.l . . . 4  |-  .<_  =  ( le `  K )
3 dihmeetlem9.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dihmeetlem9.j . . . 4  |-  .\/  =  ( join `  K )
5 dihmeetlem9.m . . . 4  |-  ./\  =  ( meet `  K )
6 dihmeetlem9.a . . . 4  |-  A  =  ( Atoms `  K )
7 dihmeetlem9.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
8 dihmeetlem9.s . . . 4  |-  .(+)  =  (
LSSum `  U )
9 dihmeetlem9.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
101, 2, 3, 4, 5, 6, 7, 8, 9dihmeetlem10N 34801 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( I `  ( ( X  ./\  Y )  .\/  p ) )  =  ( ( I `  X )  i^i  ( I `  ( Y  .\/  p ) ) ) )
1110ineq1d 3546 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( (
I `  ( ( X  ./\  Y )  .\/  p ) )  i^i  ( I `  Y
) )  =  ( ( ( I `  X )  i^i  (
I `  ( Y  .\/  p ) ) )  i^i  ( I `  Y ) ) )
12 inass 3555 . . 3  |-  ( ( ( I `  X
)  i^i  ( I `  ( Y  .\/  p
) ) )  i^i  ( I `  Y
) )  =  ( ( I `  X
)  i^i  ( (
I `  ( Y  .\/  p ) )  i^i  ( I `  Y
) ) )
13 simpl1l 1039 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  K  e.  HL )
14 hllat 32848 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1513, 14syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  K  e.  Lat )
16 simpl3 993 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  Y  e.  B )
17 simprll 761 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  p  e.  A )
181, 6atbase 32774 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
1917, 18syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  p  e.  B )
201, 2, 4latlej1 15222 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  p  e.  B )  ->  Y  .<_  ( Y  .\/  p ) )
2115, 16, 19, 20syl3anc 1218 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  Y  .<_  ( Y  .\/  p ) )
22 simpl1 991 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
231, 4latjcl 15213 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  p  e.  B )  ->  ( Y  .\/  p
)  e.  B )
2415, 16, 19, 23syl3anc 1218 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( Y  .\/  p )  e.  B
)
251, 2, 3, 9dihord 34749 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  B  /\  ( Y  .\/  p
)  e.  B )  ->  ( ( I `
 Y )  C_  ( I `  ( Y  .\/  p ) )  <-> 
Y  .<_  ( Y  .\/  p ) ) )
2622, 16, 24, 25syl3anc 1218 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( (
I `  Y )  C_  ( I `  ( Y  .\/  p ) )  <-> 
Y  .<_  ( Y  .\/  p ) ) )
2721, 26mpbird 232 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( I `  Y )  C_  (
I `  ( Y  .\/  p ) ) )
28 sseqin2 3564 . . . . 5  |-  ( ( I `  Y ) 
C_  ( I `  ( Y  .\/  p ) )  <->  ( ( I `
 ( Y  .\/  p ) )  i^i  ( I `  Y
) )  =  ( I `  Y ) )
2927, 28sylib 196 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( (
I `  ( Y  .\/  p ) )  i^i  ( I `  Y
) )  =  ( I `  Y ) )
3029ineq2d 3547 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( (
I `  X )  i^i  ( ( I `  ( Y  .\/  p ) )  i^i  ( I `
 Y ) ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
3112, 30syl5eq 2482 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( (
( I `  X
)  i^i  ( I `  ( Y  .\/  p
) ) )  i^i  ( I `  Y
) )  =  ( ( I `  X
)  i^i  ( I `  Y ) ) )
3211, 31eqtrd 2470 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( (
I `  ( ( X  ./\  Y )  .\/  p ) )  i^i  ( I `  Y
) )  =  ( ( I `  X
)  i^i  ( I `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3322    C_ wss 3323   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   meetcmee 15107   Latclat 15207   LSSumclsm 16124   Atomscatm 32748   HLchlt 32835   LHypclh 33468   DVecHcdvh 34563   DIsoHcdih 34713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-undef 6784  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-0g 14372  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-cntz 15826  df-lsm 16126  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-dvr 16763  df-drng 16812  df-lmod 16928  df-lss 16991  df-lsp 17030  df-lvec 17161  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984  df-lines 32985  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643  df-tendo 34239  df-edring 34241  df-disoa 34514  df-dvech 34564  df-dib 34624  df-dic 34658  df-dih 34714
This theorem is referenced by:  dihmeetlem12N  34803
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