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Theorem hdmaprnlem7N 31215
Description: Part of proof of part 12 in [Baer] p. 49 line 19, s-St  e. G(u'+s) = P*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hdmaprnlem1.h  |-  H  =  ( LHyp `  K
)
hdmaprnlem1.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmaprnlem1.v  |-  V  =  ( Base `  U
)
hdmaprnlem1.n  |-  N  =  ( LSpan `  U )
hdmaprnlem1.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmaprnlem1.l  |-  L  =  ( LSpan `  C )
hdmaprnlem1.m  |-  M  =  ( (mapd `  K
) `  W )
hdmaprnlem1.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmaprnlem1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmaprnlem1.se  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
hdmaprnlem1.ve  |-  ( ph  ->  v  e.  V )
hdmaprnlem1.e  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
hdmaprnlem1.ue  |-  ( ph  ->  u  e.  V )
hdmaprnlem1.un  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
hdmaprnlem1.d  |-  D  =  ( Base `  C
)
hdmaprnlem1.q  |-  Q  =  ( 0g `  C
)
hdmaprnlem1.o  |-  .0.  =  ( 0g `  U )
hdmaprnlem1.a  |-  .+b  =  ( +g  `  C )
hdmaprnlem1.t2  |-  ( ph  ->  t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
hdmaprnlem1.p  |-  .+  =  ( +g  `  U )
hdmaprnlem1.pt  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
Assertion
Ref Expression
hdmaprnlem7N  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )

Proof of Theorem hdmaprnlem7N
StepHypRef Expression
1 hdmaprnlem1.d . . 3  |-  D  =  ( Base `  C
)
2 hdmaprnlem1.a . . 3  |-  .+b  =  ( +g  `  C )
3 eqid 2258 . . 3  |-  ( -g `  C )  =  (
-g `  C )
4 hdmaprnlem1.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 hdmaprnlem1.c . . . . 5  |-  C  =  ( (LCDual `  K
) `  W )
6 hdmaprnlem1.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
74, 5, 6lcdlmod 30949 . . . 4  |-  ( ph  ->  C  e.  LMod )
8 lmodabl 15634 . . . 4  |-  ( C  e.  LMod  ->  C  e. 
Abel )
97, 8syl 17 . . 3  |-  ( ph  ->  C  e.  Abel )
10 hdmaprnlem1.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
11 hdmaprnlem1.v . . . 4  |-  V  =  ( Base `  U
)
12 hdmaprnlem1.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
13 hdmaprnlem1.ue . . . 4  |-  ( ph  ->  u  e.  V )
144, 10, 11, 5, 1, 12, 6, 13hdmapcl 31190 . . 3  |-  ( ph  ->  ( S `  u
)  e.  D )
15 hdmaprnlem1.se . . . 4  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
16 eldifi 3273 . . . 4  |-  ( s  e.  ( D  \  { Q } )  -> 
s  e.  D )
1715, 16syl 17 . . 3  |-  ( ph  ->  s  e.  D )
18 hdmaprnlem1.n . . . . 5  |-  N  =  ( LSpan `  U )
19 hdmaprnlem1.l . . . . 5  |-  L  =  ( LSpan `  C )
20 hdmaprnlem1.m . . . . 5  |-  M  =  ( (mapd `  K
) `  W )
21 hdmaprnlem1.ve . . . . 5  |-  ( ph  ->  v  e.  V )
22 hdmaprnlem1.e . . . . 5  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
23 hdmaprnlem1.un . . . . 5  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
24 hdmaprnlem1.q . . . . 5  |-  Q  =  ( 0g `  C
)
25 hdmaprnlem1.o . . . . 5  |-  .0.  =  ( 0g `  U )
26 hdmaprnlem1.t2 . . . . 5  |-  ( ph  ->  t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
274, 10, 11, 18, 5, 19, 20, 12, 6, 15, 21, 22, 13, 23, 1, 24, 25, 2, 26hdmaprnlem4tN 31212 . . . 4  |-  ( ph  ->  t  e.  V )
284, 10, 11, 5, 1, 12, 6, 27hdmapcl 31190 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
291, 2, 3, 9, 14, 17, 28, 9, 14, 17, 28ablpnpcan 15083 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  =  ( s (
-g `  C )
( S `  t
) ) )
301, 2lmodvacl 15603 . . . . 5  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
317, 14, 17, 30syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
32 eqid 2258 . . . . 5  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
331, 32, 19lspsncl 15696 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
347, 31, 33syl2anc 645 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
351, 19lspsnid 15712 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  ( L `  {
( ( S `  u )  .+b  s
) } ) )
367, 31, 35syl2anc 645 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
371, 2lmodvacl 15603 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  ( S `  t )  e.  D )  ->  (
( S `  u
)  .+b  ( S `  t ) )  e.  D )
387, 14, 28, 37syl3anc 1187 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  D )
391, 19lspsnid 15712 . . . . 5  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  ( S `  t ) )  e.  D )  ->  (
( S `  u
)  .+b  ( S `  t ) )  e.  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
407, 38, 39syl2anc 645 . . . 4  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  ( S `  t ) ) } ) )
41 hdmaprnlem1.p . . . . 5  |-  .+  =  ( +g  `  U )
42 hdmaprnlem1.pt . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
434, 10, 11, 18, 5, 19, 20, 12, 6, 15, 21, 22, 13, 23, 1, 24, 25, 2, 26, 41, 42hdmaprnlem6N 31214 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
4440, 43eleqtrrd 2335 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
453, 32lssvsubcl 15663 . . 3  |-  ( ( ( C  e.  LMod  /\  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)  /\  ( (
( S `  u
)  .+b  s )  e.  ( L `  {
( ( S `  u )  .+b  s
) } )  /\  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  -> 
( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
467, 34, 36, 44, 45syl22anc 1188 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
4729, 46eqeltrrd 2333 1  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    \ cdif 3124   {csn 3614   ` cfv 4673  (class class class)co 5792   Basecbs 13110   +g cplusg 13170   0gc0g 13362   -gcsg 14327   Abelcabel 15052   LModclmod 15589   LSubSpclss 15651   LSpanclspn 15690   HLchlt 28707   LHypclh 29340   DVecHcdvh 30435  LCDualclcd 30943  mapdcmpd 30981  HDMapchdma 31150
This theorem is referenced by:  hdmaprnlem9N  31217
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-ot 3624  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9933  df-z 9992  df-uz 10198  df-fz 10749  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-sca 13186  df-vsca 13187  df-0g 13366  df-mre 13450  df-mrc 13451  df-acs 13453  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-mnd 14329  df-submnd 14378  df-grp 14451  df-minusg 14452  df-sbg 14453  df-subg 14580  df-cntz 14755  df-oppg 14781  df-lsm 14909  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-dvr 15427  df-drng 15476  df-lmod 15591  df-lss 15652  df-lsp 15691  df-lvec 15818  df-lsatoms 28333  df-lshyp 28334  df-lcv 28376  df-lfl 28415  df-lkr 28443  df-ldual 28481  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tgrp 30099  df-tendo 30111  df-edring 30113  df-dveca 30359  df-disoa 30386  df-dvech 30436  df-dib 30496  df-dic 30530  df-dih 30586  df-doch 30705  df-djh 30752  df-lcdual 30944  df-mapd 30982  df-hvmap 31114  df-hdmap1 31151  df-hdmap 31152
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