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Theorem dih2dimbALTN 36443
Description: Extend dia2dim 36275 to isomorphism H. (This version combines dib2dim 36441 and dih2dimb 36442 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dih2dimb.l  |-  .<_  =  ( le `  K )
dih2dimb.j  |-  .\/  =  ( join `  K )
dih2dimb.a  |-  A  =  ( Atoms `  K )
dih2dimb.h  |-  H  =  ( LHyp `  K
)
dih2dimb.u  |-  U  =  ( ( DVecH `  K
) `  W )
dih2dimb.s  |-  .(+)  =  (
LSSum `  U )
dih2dimb.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dih2dimb.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dih2dimb.p  |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )
dih2dimb.q  |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )
Assertion
Ref Expression
dih2dimbALTN  |-  ( ph  ->  ( I `  ( P  .\/  Q ) ) 
C_  ( ( I `
 P )  .(+)  ( I `  Q ) ) )

Proof of Theorem dih2dimbALTN
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dih2dimb.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dih2dimb.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2467 . . . . 5  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
42, 3dibvalrel 36361 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( ( (
DIsoB `  K ) `  W ) `  ( P  .\/  Q ) ) )
51, 4syl 16 . . 3  |-  ( ph  ->  Rel  ( ( (
DIsoB `  K ) `  W ) `  ( P  .\/  Q ) ) )
6 dih2dimb.l . . . . . . 7  |-  .<_  =  ( le `  K )
7 dih2dimb.j . . . . . . 7  |-  .\/  =  ( join `  K )
8 dih2dimb.a . . . . . . 7  |-  A  =  ( Atoms `  K )
9 eqid 2467 . . . . . . 7  |-  ( (
DVecA `  K ) `  W )  =  ( ( DVecA `  K ) `  W )
10 eqid 2467 . . . . . . 7  |-  ( LSSum `  ( ( DVecA `  K
) `  W )
)  =  ( LSSum `  ( ( DVecA `  K
) `  W )
)
11 eqid 2467 . . . . . . 7  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
12 dih2dimb.p . . . . . . 7  |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )
13 dih2dimb.q . . . . . . 7  |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )
146, 7, 8, 2, 9, 10, 11, 1, 12, 13dia2dim 36275 . . . . . 6  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  ( P  .\/  Q ) )  C_  ( ( ( (
DIsoA `  K ) `  W ) `  P
) ( LSSum `  (
( DVecA `  K ) `  W ) ) ( ( ( DIsoA `  K
) `  W ) `  Q ) ) )
1514sseld 3508 . . . . 5  |-  ( ph  ->  ( f  e.  ( ( ( DIsoA `  K
) `  W ) `  ( P  .\/  Q
) )  ->  f  e.  ( ( ( (
DIsoA `  K ) `  W ) `  P
) ( LSSum `  (
( DVecA `  K ) `  W ) ) ( ( ( DIsoA `  K
) `  W ) `  Q ) ) ) )
1615anim1d 564 . . . 4  |-  ( ph  ->  ( ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  ( P  .\/  Q ) )  /\  s  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) )  ->  (
f  e.  ( ( ( ( DIsoA `  K
) `  W ) `  P ) ( LSSum `  ( ( DVecA `  K
) `  W )
) ( ( (
DIsoA `  K ) `  W ) `  Q
) )  /\  s  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) ) ) )
171simpld 459 . . . . . 6  |-  ( ph  ->  K  e.  HL )
1812simpld 459 . . . . . 6  |-  ( ph  ->  P  e.  A )
1913simpld 459 . . . . . 6  |-  ( ph  ->  Q  e.  A )
20 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2120, 7, 8hlatjcl 34564 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2217, 18, 19, 21syl3anc 1228 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2312simprd 463 . . . . . 6  |-  ( ph  ->  P  .<_  W )
2413simprd 463 . . . . . 6  |-  ( ph  ->  Q  .<_  W )
25 hllat 34561 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2617, 25syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
2720, 8atbase 34487 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2818, 27syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  ( Base `  K ) )
2920, 8atbase 34487 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3019, 29syl 16 . . . . . . 7  |-  ( ph  ->  Q  e.  ( Base `  K ) )
311simprd 463 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
3220, 2lhpbase 35195 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3331, 32syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  ( Base `  K ) )
3420, 6, 7latjle12 15566 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  W  /\  Q  .<_  W )  <-> 
( P  .\/  Q
)  .<_  W ) )
3526, 28, 30, 33, 34syl13anc 1230 . . . . . 6  |-  ( ph  ->  ( ( P  .<_  W  /\  Q  .<_  W )  <-> 
( P  .\/  Q
)  .<_  W ) )
3623, 24, 35mpbi2and 919 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  .<_  W )
37 eqid 2467 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
38 eqid 2467 . . . . . 6  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
3920, 6, 2, 37, 38, 11, 3dibopelval2 36343 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P 
.\/  Q )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  .<_  W ) )  ->  ( <. f ,  s >.  e.  ( ( ( DIsoB `  K
) `  W ) `  ( P  .\/  Q
) )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  ( P  .\/  Q ) )  /\  s  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) ) ) )
401, 22, 36, 39syl12anc 1226 . . . 4  |-  ( ph  ->  ( <. f ,  s
>.  e.  ( ( (
DIsoB `  K ) `  W ) `  ( P  .\/  Q ) )  <-> 
( f  e.  ( ( ( DIsoA `  K
) `  W ) `  ( P  .\/  Q
) )  /\  s  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) ) ) )
41 dih2dimb.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
42 dih2dimb.s . . . . 5  |-  .(+)  =  (
LSSum `  U )
4328, 23jca 532 . . . . 5  |-  ( ph  ->  ( P  e.  (
Base `  K )  /\  P  .<_  W ) )
4430, 24jca 532 . . . . 5  |-  ( ph  ->  ( Q  e.  (
Base `  K )  /\  Q  .<_  W ) )
4520, 6, 2, 37, 38, 9, 41, 10, 42, 11, 3, 1, 43, 44diblsmopel 36369 . . . 4  |-  ( ph  ->  ( <. f ,  s
>.  e.  ( ( ( ( DIsoB `  K ) `  W ) `  P
)  .(+)  ( ( (
DIsoB `  K ) `  W ) `  Q
) )  <->  ( f  e.  ( ( ( (
DIsoA `  K ) `  W ) `  P
) ( LSSum `  (
( DVecA `  K ) `  W ) ) ( ( ( DIsoA `  K
) `  W ) `  Q ) )  /\  s  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) ) ) )
4616, 40, 453imtr4d 268 . . 3  |-  ( ph  ->  ( <. f ,  s
>.  e.  ( ( (
DIsoB `  K ) `  W ) `  ( P  .\/  Q ) )  ->  <. f ,  s
>.  e.  ( ( ( ( DIsoB `  K ) `  W ) `  P
)  .(+)  ( ( (
DIsoB `  K ) `  W ) `  Q
) ) ) )
475, 46relssdv 5101 . 2  |-  ( ph  ->  ( ( ( DIsoB `  K ) `  W
) `  ( P  .\/  Q ) )  C_  ( ( ( (
DIsoB `  K ) `  W ) `  P
)  .(+)  ( ( (
DIsoB `  K ) `  W ) `  Q
) ) )
48 dih2dimb.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
4920, 6, 2, 48, 3dihvalb 36435 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P 
.\/  Q )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  .<_  W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( ( DIsoB `  K
) `  W ) `  ( P  .\/  Q
) ) )
501, 22, 36, 49syl12anc 1226 . 2  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( (
DIsoB `  K ) `  W ) `  ( P  .\/  Q ) ) )
5120, 6, 2, 48, 3dihvalb 36435 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  ( Base `  K
)  /\  P  .<_  W ) )  ->  (
I `  P )  =  ( ( (
DIsoB `  K ) `  W ) `  P
) )
521, 28, 23, 51syl12anc 1226 . . 3  |-  ( ph  ->  ( I `  P
)  =  ( ( ( DIsoB `  K ) `  W ) `  P
) )
5320, 6, 2, 48, 3dihvalb 36435 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  ( Base `  K
)  /\  Q  .<_  W ) )  ->  (
I `  Q )  =  ( ( (
DIsoB `  K ) `  W ) `  Q
) )
541, 30, 24, 53syl12anc 1226 . . 3  |-  ( ph  ->  ( I `  Q
)  =  ( ( ( DIsoB `  K ) `  W ) `  Q
) )
5552, 54oveq12d 6313 . 2  |-  ( ph  ->  ( ( I `  P )  .(+)  ( I `
 Q ) )  =  ( ( ( ( DIsoB `  K ) `  W ) `  P
)  .(+)  ( ( (
DIsoB `  K ) `  W ) `  Q
) ) )
5647, 50, 553sstr4d 3552 1  |-  ( ph  ->  ( I `  ( P  .\/  Q ) ) 
C_  ( ( I `
 P )  .(+)  ( I `  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   <.cop 4039   class class class wbr 4453    |-> cmpt 4511    _I cid 4796    |` cres 5007   Rel wrel 5010   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   Latclat 15549   LSSumclsm 16527   Atomscatm 34461   HLchlt 34548   LHypclh 35181   LTrncltrn 35298   DVecAcdveca 36199   DIsoAcdia 36226   DVecHcdvh 36276   DIsoBcdib 36336   DIsoHcdih 36426
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34157
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tgrp 35940  df-tendo 35952  df-edring 35954  df-dveca 36200  df-disoa 36227  df-dvech 36277  df-dib 36337  df-dih 36427
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator