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Theorem dihjatc1 31319
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 985 . . 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 1066 . . . . 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 29371 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . . 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 986 . . . . 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 987 . . . . 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 14206 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
104, 5, 6, 9syl3anc 1182 . . . 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 981 . . . . 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 29297 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1411, 13syl 15 . . . 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 14205 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  Q  e.  B )  ->  (
( X  ./\  Y
)  .\/  Q )  e.  B )
174, 10, 14, 16syl3anc 1182 . . 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 956 . . . . 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 983 . . . . 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 31317 . . . . 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 1190 . . . 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 31316 . . . . . 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 1190 . . . . 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 4070 . . . 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 291 . . 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 14216 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  Q  e.  B )  ->  Q  .<_  ( ( X  ./\  Y )  .\/  Q ) )
294, 10, 14, 28syl3anc 1182 . . 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 31255 . . 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 1191 . 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 2316 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
3620, 8, 35, 12, 21lhpmat 30037 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  ./\  W
)  =  ( 0.
`  K ) )
371, 18, 36syl2anc 642 . . . . . 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 5916 . . . . 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 1067 . . . . . . 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 30005 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
4139, 40syl 15 . . . . . 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 984 . . . . . 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 29866 . . . . . 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 1195 . . . . 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 29369 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
462, 45syl 15 . . . . . 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 29233 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  ./\  Y )  e.  B )  -> 
( ( X  ./\  Y )  .\/  ( 0.
`  K ) )  =  ( X  ./\  Y ) )
4846, 10, 47syl2anc 642 . . . . 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 2356 . . . 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 5567 . . 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 5916 . 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 2348 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   meetcmee 14128   0.cp0 14192   Latclat 14200   LSSumclsm 14994   OLcol 29182   Atomscatm 29271   HLchlt 29358   LHypclh 29991   DVecHcdvh 31086   DIsoHcdih 31236
This theorem is referenced by:  dihjatc2N  31320  dihjatc3  31321  dihmeetlem8N  31322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-undef 6340  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-0g 13453  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-cntz 14842  df-lsm 14996  df-cmn 15140  df-abl 15141  df-mgp 15375  df-rng 15389  df-ur 15391  df-oppr 15454  df-dvdsr 15472  df-unit 15473  df-invr 15503  df-dvr 15514  df-drng 15563  df-lmod 15678  df-lss 15739  df-lsp 15778  df-lvec 15905  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507  df-lines 29508  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166  df-tendo 30762  df-edring 30764  df-disoa 31037  df-dvech 31087  df-dib 31147  df-dic 31181  df-dih 31237
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