Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihjatc1 Structured version   Unicode version

Theorem dihjatc1 36117
Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change  .\/ order of  ( X  ./\  Y )  .\/  Q here and down? (Contributed by NM, 6-Apr-2014.)
Hypotheses
Ref Expression
dihjatc1.b  |-  B  =  ( Base `  K
)
dihjatc1.l  |-  .<_  =  ( le `  K )
dihjatc1.h  |-  H  =  ( LHyp `  K
)
dihjatc1.j  |-  .\/  =  ( join `  K )
dihjatc1.m  |-  ./\  =  ( meet `  K )
dihjatc1.a  |-  A  =  ( Atoms `  K )
dihjatc1.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjatc1.s  |-  .(+)  =  (
LSSum `  U )
dihjatc1.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihjatc1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( I `  ( ( X  ./\  Y )  .\/  Q ) )  =  ( ( I `  Q ) 
.(+)  ( I `  ( X  ./\  Y ) ) ) )

Proof of Theorem dihjatc1
StepHypRef Expression
1 simp11 1026 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp11l 1107 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  K  e.  HL )
3 hllat 34169 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  K  e.  Lat )
5 simp12 1027 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  X  e.  B )
6 simp13 1028 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  Y  e.  B )
7 dihjatc1.b . . . . . 6  |-  B  =  ( Base `  K
)
8 dihjatc1.m . . . . . 6  |-  ./\  =  ( meet `  K )
97, 8latmcl 15538 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
104, 5, 6, 9syl3anc 1228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( X  ./\ 
Y )  e.  B
)
11 simp2l 1022 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  Q  e.  A )
12 dihjatc1.a . . . . . 6  |-  A  =  ( Atoms `  K )
137, 12atbase 34095 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1411, 13syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  Q  e.  B )
15 dihjatc1.j . . . . 5  |-  .\/  =  ( join `  K )
167, 15latjcl 15537 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  Q  e.  B )  ->  (
( X  ./\  Y
)  .\/  Q )  e.  B )
174, 10, 14, 16syl3anc 1228 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( ( X  ./\  Y )  .\/  Q )  e.  B )
18 simp2 997 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
19 simp3l 1024 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  Q  .<_  X )
20 dihjatc1.l . . . . . 6  |-  .<_  =  ( le `  K )
21 dihjatc1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
227, 20, 21, 15, 8, 12dihmeetlem6 36115 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  -.  ( X  ./\  ( Y  .\/  Q ) )  .<_  W )
231, 5, 6, 18, 19, 22syl32anc 1236 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  -.  ( X  ./\  ( Y  .\/  Q ) )  .<_  W )
247, 20, 15, 8, 12dihmeetlem5 36114 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  Q
) )  =  ( ( X  ./\  Y
)  .\/  Q )
)
252, 5, 6, 11, 19, 24syl32anc 1236 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( X  ./\  ( Y  .\/  Q
) )  =  ( ( X  ./\  Y
)  .\/  Q )
)
2625breq1d 4457 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( ( X  ./\  ( Y  .\/  Q ) )  .<_  W  <->  ( ( X  ./\  Y )  .\/  Q )  .<_  W )
)
2723, 26mtbid 300 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  -.  (
( X  ./\  Y
)  .\/  Q )  .<_  W )
287, 20, 15latlej2 15547 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  Q  e.  B )  ->  Q  .<_  ( ( X  ./\  Y )  .\/  Q ) )
294, 10, 14, 28syl3anc 1228 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  Q  .<_  ( ( X  ./\  Y
)  .\/  Q )
)
30 dihjatc1.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
31 dihjatc1.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
32 dihjatc1.s . . . 4  |-  .(+)  =  (
LSSum `  U )
337, 20, 15, 8, 12, 21, 30, 31, 32dihvalcq2 36053 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( X  ./\  Y )  .\/  Q )  e.  B  /\  -.  ( ( X 
./\  Y )  .\/  Q )  .<_  W )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  ( ( X  ./\  Y )  .\/  Q ) ) )  ->  (
I `  ( ( X  ./\  Y )  .\/  Q ) )  =  ( ( I `  Q
)  .(+)  ( I `  ( ( ( X 
./\  Y )  .\/  Q )  ./\  W )
) ) )
341, 17, 27, 18, 29, 33syl122anc 1237 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( I `  ( ( X  ./\  Y )  .\/  Q ) )  =  ( ( I `  Q ) 
.(+)  ( I `  ( ( ( X 
./\  Y )  .\/  Q )  ./\  W )
) ) )
35 eqid 2467 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
3620, 8, 35, 12, 21lhpmat 34835 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  ./\  W
)  =  ( 0.
`  K ) )
371, 18, 36syl2anc 661 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( Q  ./\ 
W )  =  ( 0. `  K ) )
3837oveq2d 6299 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( ( X  ./\  Y )  .\/  ( Q  ./\  W ) )  =  ( ( X  ./\  Y )  .\/  ( 0. `  K
) ) )
39 simp11r 1108 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  W  e.  H )
407, 21lhpbase 34803 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
4139, 40syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  W  e.  B )
42 simp3r 1025 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( X  ./\ 
Y )  .<_  W )
437, 20, 15, 8, 12atmod1i2 34664 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( X  ./\  Y
)  e.  B  /\  W  e.  B )  /\  ( X  ./\  Y
)  .<_  W )  -> 
( ( X  ./\  Y )  .\/  ( Q 
./\  W ) )  =  ( ( ( X  ./\  Y )  .\/  Q )  ./\  W
) )
442, 11, 10, 41, 42, 43syl131anc 1241 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( ( X  ./\  Y )  .\/  ( Q  ./\  W ) )  =  ( ( ( X  ./\  Y
)  .\/  Q )  ./\  W ) )
45 hlol 34167 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
462, 45syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  K  e.  OL )
477, 15, 35olj01 34031 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  ./\  Y )  e.  B )  -> 
( ( X  ./\  Y )  .\/  ( 0.
`  K ) )  =  ( X  ./\  Y ) )
4846, 10, 47syl2anc 661 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( ( X  ./\  Y )  .\/  ( 0. `  K ) )  =  ( X 
./\  Y ) )
4938, 44, 483eqtr3d 2516 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( (
( X  ./\  Y
)  .\/  Q )  ./\  W )  =  ( X  ./\  Y )
)
5049fveq2d 5869 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( I `  ( ( ( X 
./\  Y )  .\/  Q )  ./\  W )
)  =  ( I `
 ( X  ./\  Y ) ) )
5150oveq2d 6299 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( (
I `  Q )  .(+)  ( I `  (
( ( X  ./\  Y )  .\/  Q ) 
./\  W ) ) )  =  ( ( I `  Q ) 
.(+)  ( I `  ( X  ./\  Y ) ) ) )
5234, 51eqtrd 2508 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .<_  X  /\  ( X  ./\  Y )  .<_  W )
)  ->  ( I `  ( ( X  ./\  Y )  .\/  Q ) )  =  ( ( I `  Q ) 
.(+)  ( I `  ( X  ./\  Y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   Basecbs 14489   lecple 14561   joincjn 15430   meetcmee 15431   0.cp0 15523   Latclat 15531   LSSumclsm 16457   OLcol 33980   Atomscatm 34069   HLchlt 34156   LHypclh 34789   DVecHcdvh 35884   DIsoHcdih 36034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-riotaBAD 33765
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-tpos 6955  df-undef 7002  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-sca 14570  df-vsca 14571  df-0g 14696  df-poset 15432  df-plt 15444  df-lub 15460  df-glb 15461  df-join 15462  df-meet 15463  df-p0 15525  df-p1 15526  df-lat 15532  df-clat 15594  df-mnd 15731  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-subg 16000  df-cntz 16157  df-lsm 16459  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-rng 16997  df-oppr 17068  df-dvdsr 17086  df-unit 17087  df-invr 17117  df-dvr 17128  df-drng 17193  df-lmod 17309  df-lss 17374  df-lsp 17413  df-lvec 17544  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304  df-lvols 34305  df-lines 34306  df-psubsp 34308  df-pmap 34309  df-padd 34601  df-lhyp 34793  df-laut 34794  df-ldil 34909  df-ltrn 34910  df-trl 34964  df-tendo 35560  df-edring 35562  df-disoa 35835  df-dvech 35885  df-dib 35945  df-dic 35979  df-dih 36035
This theorem is referenced by:  dihjatc2N  36118  dihjatc3  36119  dihmeetlem8N  36120
  Copyright terms: Public domain W3C validator