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Theorem hdmaprnlem3N 31194
Description: Part of proof of part 12 in [Baer] p. 49 line 15, T  =/= P. Our  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) is Baer's P, where P* = G(u'+s). (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 )
Assertion
Ref Expression
hdmaprnlem3N  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )

Proof of Theorem hdmaprnlem3N
StepHypRef Expression
1 hdmaprnlem1.d . . . . 5  |-  D  =  ( Base `  C
)
2 hdmaprnlem1.l . . . . 5  |-  L  =  ( LSpan `  C )
3 hdmaprnlem1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
4 hdmaprnlem1.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
5 hdmaprnlem1.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
63, 4, 5lcdlmod 30933 . . . . 5  |-  ( ph  ->  C  e.  LMod )
7 hdmaprnlem1.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
8 hdmaprnlem1.v . . . . . . 7  |-  V  =  ( Base `  U
)
9 hdmaprnlem1.s . . . . . . 7  |-  S  =  ( (HDMap `  K
) `  W )
10 hdmaprnlem1.ue . . . . . . 7  |-  ( ph  ->  u  e.  V )
113, 7, 8, 4, 1, 9, 5, 10hdmapcl 31174 . . . . . 6  |-  ( ph  ->  ( S `  u
)  e.  D )
12 hdmaprnlem1.se . . . . . . 7  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
13 eldifi 3259 . . . . . . 7  |-  ( s  e.  ( D  \  { Q } )  -> 
s  e.  D )
1412, 13syl 17 . . . . . 6  |-  ( ph  ->  s  e.  D )
15 hdmaprnlem1.a . . . . . . 7  |-  .+b  =  ( +g  `  C )
161, 15lmodvacl 15589 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
176, 11, 14, 16syl3anc 1187 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
18 eqid 2256 . . . . . 6  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
191, 18, 2lspsncl 15682 . . . . . . 7  |-  ( ( C  e.  LMod  /\  s  e.  D )  ->  ( L `  { s } )  e.  (
LSubSp `  C ) )
206, 14, 19syl2anc 645 . . . . . 6  |-  ( ph  ->  ( L `  {
s } )  e.  ( LSubSp `  C )
)
211, 2lspsnid 15698 . . . . . . 7  |-  ( ( C  e.  LMod  /\  s  e.  D )  ->  s  e.  ( L `  {
s } ) )
226, 14, 21syl2anc 645 . . . . . 6  |-  ( ph  ->  s  e.  ( L `
 { s } ) )
23 hdmaprnlem1.q . . . . . . 7  |-  Q  =  ( 0g `  C
)
243, 4, 5lcdlvec 30932 . . . . . . 7  |-  ( ph  ->  C  e.  LVec )
25 hdmaprnlem1.o . . . . . . . 8  |-  .0.  =  ( 0g `  U )
26 eqid 2256 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
273, 7, 5dvhlmod 30451 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
28 hdmaprnlem1.ve . . . . . . . . . 10  |-  ( ph  ->  v  e.  V )
29 hdmaprnlem1.n . . . . . . . . . . 11  |-  N  =  ( LSpan `  U )
308, 26, 29lspsncl 15682 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  v  e.  V )  ->  ( N `  { v } )  e.  (
LSubSp `  U ) )
3127, 28, 30syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
v } )  e.  ( LSubSp `  U )
)
32 hdmaprnlem1.un . . . . . . . . 9  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
338, 25, 26, 27, 31, 10, 32lssneln0 15657 . . . . . . . 8  |-  ( ph  ->  u  e.  ( V 
\  {  .0.  }
) )
343, 7, 8, 25, 4, 23, 1, 9, 5, 33hdmapnzcl 31189 . . . . . . 7  |-  ( ph  ->  ( S `  u
)  e.  ( D 
\  { Q }
) )
35 hdmaprnlem1.m . . . . . . . 8  |-  M  =  ( (mapd `  K
) `  W )
36 hdmaprnlem1.e . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
373, 7, 8, 29, 4, 2, 35, 9, 5, 12, 28, 36, 10, 32hdmaprnlem1N 31193 . . . . . . 7  |-  ( ph  ->  ( L `  {
( S `  u
) } )  =/=  ( L `  {
s } ) )
381, 23, 2, 24, 34, 14, 37lspsnne1 15818 . . . . . 6  |-  ( ph  ->  -.  ( S `  u )  e.  ( L `  { s } ) )
391, 15, 18, 6, 20, 22, 11, 38lssvancl2 15651 . . . . 5  |-  ( ph  ->  -.  ( ( S `
 u )  .+b  s )  e.  ( L `  { s } ) )
401, 2, 6, 17, 14, 39lspsnne2 15819 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =/=  ( L `  {
s } ) )
4140necomd 2502 . . 3  |-  ( ph  ->  ( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) )
421, 18, 2lspsncl 15682 . . . . . 6  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
436, 17, 42syl2anc 645 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
443, 35, 4, 18, 5mapdrn2 30992 . . . . 5  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
4543, 44eleqtrrd 2333 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
463, 35, 5, 45mapdcnvid2 30998 . . 3  |-  ( ph  ->  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  =  ( L `  {
( ( S `  u )  .+b  s
) } ) )
4741, 36, 463netr4d 2446 . 2  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =/=  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) ) )
483, 35, 7, 26, 5, 45mapdcnvcl 30993 . . . 4  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  e.  ( LSubSp `  U )
)
493, 7, 26, 35, 5, 31, 48mapd11 30980 . . 3  |-  ( ph  ->  ( ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  <->  ( N `  { v } )  =  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) ) )
5049necon3bid 2454 . 2  |-  ( ph  ->  ( ( M `  ( N `  { v } ) )  =/=  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  <->  ( N `  { v } )  =/=  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) ) )
5147, 50mpbid 203 1  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419    \ cdif 3110   {csn 3600   `'ccnv 4646   ran crn 4648   ` cfv 4659  (class class class)co 5778   Basecbs 13096   +g cplusg 13156   0gc0g 13348   LModclmod 15575   LSubSpclss 15637   LSpanclspn 15676   HLchlt 28691   LHypclh 29324   DVecHcdvh 30419  LCDualclcd 30927  mapdcmpd 30965  HDMapchdma 31134
This theorem is referenced by:  hdmaprnlem9N  31201  hdmaprnlem3eN  31202
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 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-ot 3610  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-tpos 6154  df-iota 6211  df-undef 6250  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-n0 9919  df-z 9978  df-uz 10184  df-fz 10735  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-sca 13172  df-vsca 13173  df-0g 13352  df-mre 13436  df-mrc 13437  df-acs 13439  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-mnd 14315  df-submnd 14364  df-grp 14437  df-minusg 14438  df-sbg 14439  df-subg 14566  df-cntz 14741  df-oppg 14767  df-lsm 14895  df-cmn 15039  df-abl 15040  df-mgp 15274  df-ring 15288  df-ur 15290  df-oppr 15353  df-dvdsr 15371  df-unit 15372  df-invr 15402  df-dvr 15413  df-drng 15462  df-lmod 15577  df-lss 15638  df-lsp 15677  df-lvec 15804  df-lsatoms 28317  df-lshyp 28318  df-lcv 28360  df-lfl 28399  df-lkr 28427  df-ldual 28465  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499  df-tgrp 30083  df-tendo 30095  df-edring 30097  df-dveca 30343  df-disoa 30370  df-dvech 30420  df-dib 30480  df-dic 30514  df-dih 30570  df-doch 30689  df-djh 30736  df-lcdual 30928  df-mapd 30966  df-hvmap 31098  df-hdmap1 31135  df-hdmap 31136
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