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Theorem hdmaprnlem7N 30737
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 2253 . . 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 30471 . . . 4  |-  ( ph  ->  C  e.  LMod )
8 lmodabl 15507 . . . 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 30712 . . 3  |-  ( ph  ->  ( S `  u
)  e.  D )
15 hdmaprnlem1.se . . . 4  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
16 eldifi 3215 . . . 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 30734 . . . 4  |-  ( ph  ->  t  e.  V )
284, 10, 11, 5, 1, 12, 6, 27hdmapcl 30712 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
291, 2, 3, 9, 14, 17, 28, 9, 14, 17, 28ablpnpcan 14956 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  =  ( s (
-g `  C )
( S `  t
) ) )
301, 2lmodvacl 15476 . . . . 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 2253 . . . . 5  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
331, 32, 19lspsncl 15569 . . . 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 15585 . . . 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 15476 . . . . . 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 15585 . . . . 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 30736 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
4440, 43eleqtrrd 2330 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
453, 32lssvsubcl 15536 . . 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 2328 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 3075   {csn 3544   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   0gc0g 13274   -gcsg 14200   Abelcabel 14925   LModclmod 15462   LSubSpclss 15524   LSpanclspn 15563   HLchlt 28229   LHypclh 28862   DVecHcdvh 29957  LCDualclcd 30465  mapdcmpd 30503  HDMapchdma 30672
This theorem is referenced by:  hdmaprnlem9N  30739
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-ot 3554  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-oppg 14654  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 27855  df-lshyp 27856  df-lcv 27898  df-lfl 27937  df-lkr 27965  df-ldual 28003  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037  df-tgrp 29621  df-tendo 29633  df-edring 29635  df-dveca 29881  df-disoa 29908  df-dvech 29958  df-dib 30018  df-dic 30052  df-dih 30108  df-doch 30227  df-djh 30274  df-lcdual 30466  df-mapd 30504  df-hvmap 30636  df-hdmap1 30673  df-hdmap 30674
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