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Theorem dihjatc1 37451
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 1024 . . 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 1105 . . . . 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 35501 . . . . 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 1025 . . . . 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 1026 . . . . 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 15799 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
104, 5, 6, 9syl3anc 1226 . . . 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 1020 . . . . 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 35427 . . . . 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 15798 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  Q  e.  B )  ->  (
( X  ./\  Y
)  .\/  Q )  e.  B )
174, 10, 14, 16syl3anc 1226 . . 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 995 . . . . 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 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  .<_  X )
20 dihjatc1.l . . . . . 6  |-  .<_  =  ( le `  K )
21 dihjatc1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
227, 20, 21, 15, 8, 12dihmeetlem6 37449 . . . . 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 1234 . . . 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 37448 . . . . . 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 1234 . . . . 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 4377 . . . 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 298 . . 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 15808 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  Q  e.  B )  ->  Q  .<_  ( ( X  ./\  Y )  .\/  Q ) )
294, 10, 14, 28syl3anc 1226 . . 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 37387 . . 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 1235 . 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 2382 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
3620, 8, 35, 12, 21lhpmat 36167 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  ./\  W
)  =  ( 0.
`  K ) )
371, 18, 36syl2anc 659 . . . . . 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 6212 . . . . 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 1106 . . . . . . 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 36135 . . . . . . 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 1023 . . . . . 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 35996 . . . . . 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 1239 . . . . 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 35499 . . . . . . 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 35363 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  ./\  Y )  e.  B )  -> 
( ( X  ./\  Y )  .\/  ( 0.
`  K ) )  =  ( X  ./\  Y ) )
4846, 10, 47syl2anc 659 . . . . 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 2431 . . . 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 5778 . . 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 6212 . 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 2423 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   meetcmee 15691   0.cp0 15784   Latclat 15792   LSSumclsm 16771   OLcol 35312   Atomscatm 35401   HLchlt 35488   LHypclh 36121   DVecHcdvh 37218   DIsoHcdih 37368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-riotaBAD 35097
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-tpos 6873  df-undef 6920  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-sca 14718  df-vsca 14719  df-0g 14849  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-p1 15787  df-lat 15793  df-clat 15855  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-grp 16174  df-minusg 16175  df-sbg 16176  df-subg 16315  df-cntz 16472  df-lsm 16773  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-ring 17313  df-oppr 17385  df-dvdsr 17403  df-unit 17404  df-invr 17434  df-dvr 17445  df-drng 17511  df-lmod 17627  df-lss 17692  df-lsp 17731  df-lvec 17862  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489  df-llines 35635  df-lplanes 35636  df-lvols 35637  df-lines 35638  df-psubsp 35640  df-pmap 35641  df-padd 35933  df-lhyp 36125  df-laut 36126  df-ldil 36241  df-ltrn 36242  df-trl 36297  df-tendo 36894  df-edring 36896  df-disoa 37169  df-dvech 37219  df-dib 37279  df-dic 37313  df-dih 37369
This theorem is referenced by:  dihjatc2N  37452  dihjatc3  37453  dihmeetlem8N  37454
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