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Theorem hdmaprnlem3uN 31174
Description: Part of proof of part 12 in [Baer] p. 49. (Contributed by NM, 29-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
hdmaprnlem3uN  |-  ( ph  ->  ( N `  {
u } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )

Proof of Theorem hdmaprnlem3uN
StepHypRef Expression
1 hdmaprnlem1.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmaprnlem1.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
3 hdmaprnlem1.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
4 eqid 2256 . . 3  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmaprnlem1.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 30430 . . . 4  |-  ( ph  ->  U  e.  LMod )
7 hdmaprnlem1.ue . . . 4  |-  ( ph  ->  u  e.  V )
8 hdmaprnlem1.v . . . . 5  |-  V  =  ( Base `  U
)
9 hdmaprnlem1.n . . . . 5  |-  N  =  ( LSpan `  U )
108, 4, 9lspsncl 15661 . . . 4  |-  ( ( U  e.  LMod  /\  u  e.  V )  ->  ( N `  { u } )  e.  (
LSubSp `  U ) )
116, 7, 10syl2anc 645 . . 3  |-  ( ph  ->  ( N `  {
u } )  e.  ( LSubSp `  U )
)
121, 2, 3, 4, 5, 11mapdcnvid1N 30974 . 2  |-  ( ph  ->  ( `' M `  ( M `  ( N `
 { u }
) ) )  =  ( N `  {
u } ) )
13 hdmaprnlem1.c . . . . 5  |-  C  =  ( (LCDual `  K
) `  W )
14 hdmaprnlem1.l . . . . 5  |-  L  =  ( LSpan `  C )
15 hdmaprnlem1.s . . . . 5  |-  S  =  ( (HDMap `  K
) `  W )
161, 3, 8, 9, 13, 14, 2, 15, 5, 7hdmap10 31163 . . . 4  |-  ( ph  ->  ( M `  ( N `  { u } ) )  =  ( L `  {
( S `  u
) } ) )
17 hdmaprnlem1.d . . . . 5  |-  D  =  ( Base `  C
)
18 hdmaprnlem1.a . . . . 5  |-  .+b  =  ( +g  `  C )
19 hdmaprnlem1.q . . . . 5  |-  Q  =  ( 0g `  C
)
201, 13, 5lcdlvec 30911 . . . . 5  |-  ( ph  ->  C  e.  LVec )
211, 3, 8, 13, 17, 15, 5, 7hdmapcl 31153 . . . . 5  |-  ( ph  ->  ( S `  u
)  e.  D )
22 hdmaprnlem1.se . . . . 5  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
23 hdmaprnlem1.ve . . . . . 6  |-  ( ph  ->  v  e.  V )
24 hdmaprnlem1.e . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
25 hdmaprnlem1.un . . . . . 6  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
261, 3, 8, 9, 13, 14, 2, 15, 5, 22, 23, 24, 7, 25hdmaprnlem1N 31172 . . . . 5  |-  ( ph  ->  ( L `  {
( S `  u
) } )  =/=  ( L `  {
s } ) )
2717, 18, 19, 14, 20, 21, 22, 26lspindp3 15816 . . . 4  |-  ( ph  ->  ( L `  {
( S `  u
) } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) )
2816, 27eqnetrd 2437 . . 3  |-  ( ph  ->  ( M `  ( N `  { u } ) )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) )
291, 2, 3, 4, 5, 11mapdcl 30973 . . . . 5  |-  ( ph  ->  ( M `  ( N `  { u } ) )  e. 
ran  M )
301, 13, 5lcdlmod 30912 . . . . . . 7  |-  ( ph  ->  C  e.  LMod )
31 eldifi 3240 . . . . . . . . 9  |-  ( s  e.  ( D  \  { Q } )  -> 
s  e.  D )
3222, 31syl 17 . . . . . . . 8  |-  ( ph  ->  s  e.  D )
3317, 18lmodvacl 15568 . . . . . . . 8  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
3430, 21, 32, 33syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
35 eqid 2256 . . . . . . . 8  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
3617, 35, 14lspsncl 15661 . . . . . . 7  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
3730, 34, 36syl2anc 645 . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
381, 2, 13, 35, 5mapdrn2 30971 . . . . . 6  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
3937, 38eleqtrrd 2333 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
401, 2, 5, 29, 39mapdcnv11N 30979 . . . 4  |-  ( ph  ->  ( ( `' M `  ( M `  ( N `  { u } ) ) )  =  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( M `  ( N `  { u } ) )  =  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
4140necon3bid 2454 . . 3  |-  ( ph  ->  ( ( `' M `  ( M `  ( N `  { u } ) ) )  =/=  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( M `  ( N `  { u } ) )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
4228, 41mpbird 225 . 2  |-  ( ph  ->  ( `' M `  ( M `  ( N `
 { u }
) ) )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
4312, 42eqnetrrd 2439 1  |-  ( ph  ->  ( N `  {
u } )  =/=  ( `' 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 3091   {csn 3581   `'ccnv 4625   ran crn 4627   ` cfv 4638  (class class class)co 5757   Basecbs 13075   +g cplusg 13135   0gc0g 13327   LModclmod 15554   LSubSpclss 15616   LSpanclspn 15655   HLchlt 28670   LHypclh 29303   DVecHcdvh 30398  LCDualclcd 30906  mapdcmpd 30944  HDMapchdma 31113
This theorem is referenced by:  hdmaprnlem3eN  31181
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-ot 3591  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-tpos 6133  df-iota 6190  df-undef 6229  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-n0 9898  df-z 9957  df-uz 10163  df-fz 10714  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-sca 13151  df-vsca 13152  df-0g 13331  df-mre 13415  df-mrc 13416  df-acs 13418  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-mnd 14294  df-submnd 14343  df-grp 14416  df-minusg 14417  df-sbg 14418  df-subg 14545  df-cntz 14720  df-oppg 14746  df-lsm 14874  df-cmn 15018  df-abl 15019  df-mgp 15253  df-ring 15267  df-ur 15269  df-oppr 15332  df-dvdsr 15350  df-unit 15351  df-invr 15381  df-dvr 15392  df-drng 15441  df-lmod 15556  df-lss 15617  df-lsp 15656  df-lvec 15783  df-lsatoms 28296  df-lshyp 28297  df-lcv 28339  df-lfl 28378  df-lkr 28406  df-ldual 28444  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478  df-tgrp 30062  df-tendo 30074  df-edring 30076  df-dveca 30322  df-disoa 30349  df-dvech 30399  df-dib 30459  df-dic 30493  df-dih 30549  df-doch 30668  df-djh 30715  df-lcdual 30907  df-mapd 30945  df-hvmap 31077  df-hdmap1 31114  df-hdmap 31115
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