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Theorem dihvalcq2 35174
Description: Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. (Contributed by NM, 26-Sep-2014.)
Hypotheses
Ref Expression
dihvalcq2.b  |-  B  =  ( Base `  K
)
dihvalcq2.l  |-  .<_  =  ( le `  K )
dihvalcq2.j  |-  .\/  =  ( join `  K )
dihvalcq2.m  |-  ./\  =  ( meet `  K )
dihvalcq2.a  |-  A  =  ( Atoms `  K )
dihvalcq2.h  |-  H  =  ( LHyp `  K
)
dihvalcq2.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihvalcq2.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihvalcq2.p  |-  .(+)  =  (
LSSum `  U )
Assertion
Ref Expression
dihvalcq2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( I `  X )  =  ( ( I `  Q
)  .(+)  ( I `  ( X  ./\  W ) ) ) )

Proof of Theorem dihvalcq2
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2 989 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
3 simp3l 1016 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp3r 1017 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  Q  .<_  X )
5 dihvalcq2.b . . . . . 6  |-  B  =  ( Base `  K
)
6 dihvalcq2.l . . . . . 6  |-  .<_  =  ( le `  K )
7 dihvalcq2.j . . . . . 6  |-  .\/  =  ( join `  K )
8 dihvalcq2.m . . . . . 6  |-  ./\  =  ( meet `  K )
9 dihvalcq2.a . . . . . 6  |-  A  =  ( Atoms `  K )
10 dihvalcq2.h . . . . . 6  |-  H  =  ( LHyp `  K
)
115, 6, 7, 8, 9, 10lhpmcvr3 33951 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( Q  .<_  X  <->  ( Q  .\/  ( X  ./\  W ) )  =  X ) )
121, 2, 3, 11syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( Q  .<_  X  <->  ( Q  .\/  ( X  ./\  W ) )  =  X ) )
134, 12mpbid 210 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( Q  .\/  ( X  ./\  W
) )  =  X )
14 dihvalcq2.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
15 eqid 2450 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
16 eqid 2450 . . . 4  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
17 dihvalcq2.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
18 dihvalcq2.p . . . 4  |-  .(+)  =  (
LSSum `  U )
195, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18dihvalcq 35163 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( I `  X )  =  ( ( ( ( DIsoC `  K ) `  W
) `  Q )  .(+)  ( ( ( DIsoB `  K ) `  W
) `  ( X  ./\ 
W ) ) ) )
201, 2, 3, 13, 19syl112anc 1223 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( I `  X )  =  ( ( ( ( DIsoC `  K ) `  W
) `  Q )  .(+)  ( ( ( DIsoB `  K ) `  W
) `  ( X  ./\ 
W ) ) ) )
216, 9, 10, 16, 14dihvalcqat 35166 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( ( ( DIsoC `  K ) `  W ) `  Q
) )
221, 3, 21syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( I `  Q )  =  ( ( ( DIsoC `  K
) `  W ) `  Q ) )
23 simp1l 1012 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  K  e.  HL )
24 hllat 33290 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2523, 24syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  K  e.  Lat )
26 simp2l 1014 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  X  e.  B )
27 simp1r 1013 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  W  e.  H )
285, 10lhpbase 33924 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2927, 28syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  W  e.  B )
305, 8latmcl 15310 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
3125, 26, 29, 30syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( X  ./\ 
W )  e.  B
)
325, 6, 8latmle2 15335 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
3325, 26, 29, 32syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( X  ./\ 
W )  .<_  W )
345, 6, 10, 14, 15dihvalb 35164 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X 
./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) )  -> 
( I `  ( X  ./\  W ) )  =  ( ( (
DIsoB `  K ) `  W ) `  ( X  ./\  W ) ) )
351, 31, 33, 34syl12anc 1217 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( I `  ( X  ./\  W
) )  =  ( ( ( DIsoB `  K
) `  W ) `  ( X  ./\  W
) ) )
3622, 35oveq12d 6194 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( (
I `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  =  ( ( ( ( DIsoC `  K
) `  W ) `  Q )  .(+)  ( ( ( DIsoB `  K ) `  W ) `  ( X  ./\  W ) ) ) )
3720, 36eqtr4d 2493 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( I `  X )  =  ( ( I `  Q
)  .(+)  ( I `  ( X  ./\  W ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   meetcmee 15203   Latclat 15303   LSSumclsm 16223   Atomscatm 33190   HLchlt 33277   LHypclh 33910   DVecHcdvh 35005   DIsoBcdib 35065   DIsoCcdic 35099   DIsoHcdih 35155
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-riotaBAD 32886
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-tpos 6831  df-undef 6878  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-sca 14342  df-vsca 14343  df-0g 14468  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-p1 15298  df-lat 15304  df-clat 15366  df-mnd 15503  df-submnd 15553  df-grp 15633  df-minusg 15634  df-sbg 15635  df-subg 15766  df-cntz 15923  df-lsm 16225  df-cmn 16369  df-abl 16370  df-mgp 16683  df-ur 16695  df-rng 16739  df-oppr 16807  df-dvdsr 16825  df-unit 16826  df-invr 16856  df-dvr 16867  df-drng 16926  df-lmod 17042  df-lss 17106  df-lsp 17145  df-lvec 17276  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425  df-lvols 33426  df-lines 33427  df-psubsp 33429  df-pmap 33430  df-padd 33722  df-lhyp 33914  df-laut 33915  df-ldil 34030  df-ltrn 34031  df-trl 34085  df-tendo 34681  df-edring 34683  df-disoa 34956  df-dvech 35006  df-dib 35066  df-dic 35100  df-dih 35156
This theorem is referenced by:  dihjatc1  35238  dihjatcclem1  35345
  Copyright terms: Public domain W3C validator