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Theorem hdmap11lem1 37294
Description: Lemma for hdmapadd 37296. (Contributed by NM, 26-May-2015.)
Hypotheses
Ref Expression
hdmap11.h  |-  H  =  ( LHyp `  K
)
hdmap11.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap11.v  |-  V  =  ( Base `  U
)
hdmap11.p  |-  .+  =  ( +g  `  U )
hdmap11.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap11.a  |-  .+b  =  ( +g  `  C )
hdmap11.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmap11.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap11.x  |-  ( ph  ->  X  e.  V )
hdmap11.y  |-  ( ph  ->  Y  e.  V )
hdmap11.e  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
hdmap11.o  |-  .0.  =  ( 0g `  U )
hdmap11.n  |-  N  =  ( LSpan `  U )
hdmap11.d  |-  D  =  ( Base `  C
)
hdmap11.l  |-  L  =  ( LSpan `  C )
hdmap11.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap11.j  |-  J  =  ( (HVMap `  K
) `  W )
hdmap11.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap11lem0.1a  |-  ( ph  ->  z  e.  V )
hdmap11lem0.6  |-  ( ph  ->  -.  z  e.  ( N `  { X ,  Y } ) )
hdmap11lem0.2a  |-  ( ph  ->  ( N `  {
z } )  =/=  ( N `  { E } ) )
Assertion
Ref Expression
hdmap11lem1  |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( ( S `
 X )  .+b  ( S `  Y ) ) )

Proof of Theorem hdmap11lem1
StepHypRef Expression
1 hdmap11.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap11.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap11.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap11.p . . 3  |-  .+  =  ( +g  `  U )
5 hdmap11.o . . 3  |-  .0.  =  ( 0g `  U )
6 hdmap11.n . . 3  |-  N  =  ( LSpan `  U )
7 hdmap11.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap11.d . . 3  |-  D  =  ( Base `  C
)
9 hdmap11.a . . 3  |-  .+b  =  ( +g  `  C )
10 hdmap11.l . . 3  |-  L  =  ( LSpan `  C )
11 hdmap11.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
12 hdmap11.i . . 3  |-  I  =  ( (HDMap1 `  K
) `  W )
13 hdmap11.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
14 eqid 2441 . . . . . 6  |-  ( 0g
`  C )  =  ( 0g `  C
)
15 hdmap11.j . . . . . 6  |-  J  =  ( (HVMap `  K
) `  W )
16 eqid 2441 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2441 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
18 hdmap11.e . . . . . . 7  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
191, 16, 17, 2, 3, 5, 18, 13dvheveccl 36562 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  {  .0.  }
) )
201, 2, 3, 5, 7, 8, 14, 15, 13, 19hvmapcl2 37216 . . . . 5  |-  ( ph  ->  ( J `  E
)  e.  ( D 
\  { ( 0g
`  C ) } ) )
2120eldifad 3471 . . . 4  |-  ( ph  ->  ( J `  E
)  e.  D )
221, 2, 3, 5, 6, 7, 10, 11, 15, 13, 19mapdhvmap 37219 . . . 4  |-  ( ph  ->  ( M `  ( N `  { E } ) )  =  ( L `  {
( J `  E
) } ) )
23 hdmap11lem0.2a . . . . 5  |-  ( ph  ->  ( N `  {
z } )  =/=  ( N `  { E } ) )
2423necomd 2712 . . . 4  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { z } ) )
25 hdmap11lem0.1a . . . 4  |-  ( ph  ->  z  e.  V )
261, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 21, 22, 24, 19, 25hdmap1cl 37255 . . 3  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  z
>. )  e.  D
)
27 eqid 2441 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
281, 2, 13dvhlmod 36560 . . . 4  |-  ( ph  ->  U  e.  LMod )
29 hdmap11.x . . . . 5  |-  ( ph  ->  X  e.  V )
30 hdmap11.y . . . . 5  |-  ( ph  ->  Y  e.  V )
313, 27, 6, 28, 29, 30lspprcl 17495 . . . 4  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
32 hdmap11lem0.6 . . . 4  |-  ( ph  ->  -.  z  e.  ( N `  { X ,  Y } ) )
333, 5, 27, 28, 31, 25, 32lssneln0 17469 . . 3  |-  ( ph  ->  z  e.  ( V 
\  {  .0.  }
) )
34 eqidd 2442 . . . . 5  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  z
>. )  =  (
I `  <. E , 
( J `  E
) ,  z >.
) )
35 eqid 2441 . . . . . 6  |-  ( -g `  U )  =  (
-g `  U )
36 eqid 2441 . . . . . 6  |-  ( -g `  C )  =  (
-g `  C )
371, 2, 3, 35, 5, 6, 7, 8, 36, 10, 11, 12, 13, 19, 21, 33, 26, 24, 22hdmap1eq 37252 . . . . 5  |-  ( ph  ->  ( ( I `  <. E ,  ( J `
 E ) ,  z >. )  =  ( I `  <. E , 
( J `  E
) ,  z >.
)  <->  ( ( M `
 ( N `  { z } ) )  =  ( L `
 { ( I `
 <. E ,  ( J `  E ) ,  z >. ) } )  /\  ( M `  ( N `  { ( E (
-g `  U )
z ) } ) )  =  ( L `
 { ( ( J `  E ) ( -g `  C
) ( I `  <. E ,  ( J `
 E ) ,  z >. ) ) } ) ) ) )
3834, 37mpbid 210 . . . 4  |-  ( ph  ->  ( ( M `  ( N `  { z } ) )  =  ( L `  {
( I `  <. E ,  ( J `  E ) ,  z
>. ) } )  /\  ( M `  ( N `
 { ( E ( -g `  U
) z ) } ) )  =  ( L `  { ( ( J `  E
) ( -g `  C
) ( I `  <. E ,  ( J `
 E ) ,  z >. ) ) } ) ) )
3938simpld 459 . . 3  |-  ( ph  ->  ( M `  ( N `  { z } ) )  =  ( L `  {
( I `  <. E ,  ( J `  E ) ,  z
>. ) } ) )
401, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 33, 29, 30, 32, 39hdmap1l6 37272 . 2  |-  ( ph  ->  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  ( X  .+  Y )
>. )  =  (
( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  X >. )  .+b  ( I `  <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  Y >. ) ) )
41 hdmap11.s . . 3  |-  S  =  ( (HDMap `  K
) `  W )
423, 4lmodvacl 17397 . . . 4  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
4328, 29, 30, 42syl3anc 1227 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
441, 2, 13dvhlvec 36559 . . . 4  |-  ( ph  ->  U  e.  LVec )
4519eldifad 3471 . . . 4  |-  ( ph  ->  E  e.  V )
463, 4, 6, 28, 29, 30lspprvacl 17516 . . . . . . 7  |-  ( ph  ->  ( X  .+  Y
)  e.  ( N `
 { X ,  Y } ) )
4727, 6, 28, 31, 46lspsnel5a 17513 . . . . . 6  |-  ( ph  ->  ( N `  {
( X  .+  Y
) } )  C_  ( N `  { X ,  Y } ) )
4847, 32ssneldd 3490 . . . . 5  |-  ( ph  ->  -.  z  e.  ( N `  { ( X  .+  Y ) } ) )
493, 6, 28, 25, 43, 48lspsnne2 17635 . . . 4  |-  ( ph  ->  ( N `  {
z } )  =/=  ( N `  {
( X  .+  Y
) } ) )
503, 6, 5, 44, 45, 43, 33, 23, 49hdmaplem4 37224 . . 3  |-  ( ph  ->  -.  z  e.  ( ( N `  { E } )  u.  ( N `  { ( X  .+  Y ) } ) ) )
511, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 43, 25, 50hdmapval2 37285 . 2  |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( I `  <. z ,  ( I `
 <. E ,  ( J `  E ) ,  z >. ) ,  ( X  .+  Y ) >. )
)
523, 6, 44, 25, 29, 30, 32lspindpi 17649 . . . . . 6  |-  ( ph  ->  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { Y } ) ) )
5352simpld 459 . . . . 5  |-  ( ph  ->  ( N `  {
z } )  =/=  ( N `  { X } ) )
543, 6, 5, 44, 45, 29, 33, 23, 53hdmaplem4 37224 . . . 4  |-  ( ph  ->  -.  z  e.  ( ( N `  { E } )  u.  ( N `  { X } ) ) )
551, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 29, 25, 54hdmapval2 37285 . . 3  |-  ( ph  ->  ( S `  X
)  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  X >. ) )
5652simprd 463 . . . . 5  |-  ( ph  ->  ( N `  {
z } )  =/=  ( N `  { Y } ) )
573, 6, 5, 44, 45, 30, 33, 23, 56hdmaplem4 37224 . . . 4  |-  ( ph  ->  -.  z  e.  ( ( N `  { E } )  u.  ( N `  { Y } ) ) )
581, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 30, 25, 57hdmapval2 37285 . . 3  |-  ( ph  ->  ( S `  Y
)  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  Y >. ) )
5955, 58oveq12d 6296 . 2  |-  ( ph  ->  ( ( S `  X )  .+b  ( S `  Y )
)  =  ( ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  X >. )  .+b  ( I `  <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  Y >. ) ) )
6040, 51, 593eqtr4d 2492 1  |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( ( S `
 X )  .+b  ( S `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   {csn 4011   {cpr 4013   <.cop 4017   <.cotp 4019    _I cid 4777    |` cres 4988   ` cfv 5575  (class class class)co 6278   Basecbs 14506   +g cplusg 14571   0gc0g 14711   -gcsg 15926   LModclmod 17383   LSubSpclss 17449   LSpanclspn 17488   HLchlt 34798   LHypclh 35431   LTrncltrn 35548   DVecHcdvh 36528  LCDualclcd 37036  mapdcmpd 37074  HVMapchvm 37206  HDMap1chdma1 37242  HDMapchdma 37243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-riotaBAD 34407
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-ot 4020  df-uni 4232  df-int 4269  df-iun 4314  df-iin 4315  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6522  df-om 6683  df-1st 6782  df-2nd 6783  df-tpos 6954  df-undef 7001  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-n0 10799  df-z 10868  df-uz 11088  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-sca 14587  df-vsca 14588  df-0g 14713  df-mre 14857  df-mrc 14858  df-acs 14860  df-preset 15428  df-poset 15446  df-plt 15459  df-lub 15475  df-glb 15476  df-join 15477  df-meet 15478  df-p0 15540  df-p1 15541  df-lat 15547  df-clat 15609  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-submnd 15838  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069  df-cntz 16226  df-oppg 16252  df-lsm 16527  df-cmn 16671  df-abl 16672  df-mgp 17013  df-ur 17025  df-ring 17071  df-oppr 17143  df-dvdsr 17161  df-unit 17162  df-invr 17192  df-dvr 17203  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lsatoms 34424  df-lshyp 34425  df-lcv 34467  df-lfl 34506  df-lkr 34534  df-ldual 34572  df-oposet 34624  df-ol 34626  df-oml 34627  df-covers 34714  df-ats 34715  df-atl 34746  df-cvlat 34770  df-hlat 34799  df-llines 34945  df-lplanes 34946  df-lvols 34947  df-lines 34948  df-psubsp 34950  df-pmap 34951  df-padd 35243  df-lhyp 35435  df-laut 35436  df-ldil 35551  df-ltrn 35552  df-trl 35607  df-tgrp 36192  df-tendo 36204  df-edring 36206  df-dveca 36452  df-disoa 36479  df-dvech 36529  df-dib 36589  df-dic 36623  df-dih 36679  df-doch 36798  df-djh 36845  df-lcdual 37037  df-mapd 37075  df-hvmap 37207  df-hdmap1 37244  df-hdmap 37245
This theorem is referenced by:  hdmap11lem2  37295
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