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Theorem hdmaprnlem9N 31317
Description: Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 31096 and mapdcnv11N 31116. Use better hypotheses and/or theorems? (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
hdmaprnlem9N  |-  ( ph  ->  s  =  ( S `
 t ) )

Proof of Theorem hdmaprnlem9N
StepHypRef Expression
1 hdmaprnlem1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 hdmaprnlem1.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmaprnlem1.v . . . . . 6  |-  V  =  ( Base `  U
)
4 hdmaprnlem1.n . . . . . 6  |-  N  =  ( LSpan `  U )
5 hdmaprnlem1.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
6 hdmaprnlem1.l . . . . . 6  |-  L  =  ( LSpan `  C )
7 hdmaprnlem1.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
8 hdmaprnlem1.s . . . . . 6  |-  S  =  ( (HDMap `  K
) `  W )
9 hdmaprnlem1.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
10 hdmaprnlem1.se . . . . . 6  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
11 hdmaprnlem1.ve . . . . . 6  |-  ( ph  ->  v  e.  V )
12 hdmaprnlem1.e . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
13 hdmaprnlem1.ue . . . . . 6  |-  ( ph  ->  u  e.  V )
14 hdmaprnlem1.un . . . . . 6  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
15 hdmaprnlem1.d . . . . . 6  |-  D  =  ( Base `  C
)
16 hdmaprnlem1.q . . . . . 6  |-  Q  =  ( 0g `  C
)
17 hdmaprnlem1.o . . . . . 6  |-  .0.  =  ( 0g `  U )
18 hdmaprnlem1.a . . . . . 6  |-  .+b  =  ( +g  `  C )
19 hdmaprnlem1.t2 . . . . . 6  |-  ( ph  ->  t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
20 hdmaprnlem1.p . . . . . 6  |-  .+  =  ( +g  `  U )
21 hdmaprnlem1.pt . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
221, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21hdmaprnlem7N 31315 . . . . 5  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21hdmaprnlem8N 31316 . . . . . 6  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( M `  ( N `  { t } ) ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19hdmaprnlem4N 31313 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { t } ) )  =  ( L `  {
s } ) )
2523, 24eleqtrd 2360 . . . . 5  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { s } ) )
26 elin 3359 . . . . 5  |-  ( ( s ( -g `  C
) ( S `  t ) )  e.  ( ( L `  { ( ( S `
 u )  .+b  s ) } )  i^i  ( L `  { s } ) )  <->  ( ( s ( -g `  C
) ( S `  t ) )  e.  ( L `  {
( ( S `  u )  .+b  s
) } )  /\  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { s } ) ) )
2722, 25, 26sylanbrc 647 . . . 4  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( ( L `
 { ( ( S `  u ) 
.+b  s ) } )  i^i  ( L `
 { s } ) ) )
281, 5, 9lcdlvec 31048 . . . . 5  |-  ( ph  ->  C  e.  LVec )
291, 5, 9lcdlmod 31049 . . . . . 6  |-  ( ph  ->  C  e.  LMod )
301, 2, 3, 5, 15, 8, 9, 13hdmapcl 31290 . . . . . 6  |-  ( ph  ->  ( S `  u
)  e.  D )
31 eldifi 3299 . . . . . . 7  |-  ( s  e.  ( D  \  { Q } )  -> 
s  e.  D )
3210, 31syl 17 . . . . . 6  |-  ( ph  ->  s  e.  D )
3315, 18lmodvacl 15635 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
3429, 30, 32, 33syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
35 eqid 2284 . . . . . . . . . . . . . 14  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
3615, 35, 6lspsncl 15728 . . . . . . . . . . . . 13  |-  ( ( C  e.  LMod  /\  s  e.  D )  ->  ( L `  { s } )  e.  (
LSubSp `  C ) )
3729, 32, 36syl2anc 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( L `  {
s } )  e.  ( LSubSp `  C )
)
381, 7, 5, 35, 9mapdrn2 31108 . . . . . . . . . . . 12  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
3937, 38eleqtrrd 2361 . . . . . . . . . . 11  |-  ( ph  ->  ( L `  {
s } )  e. 
ran  M )
401, 7, 9, 39mapdcnvid2 31114 . . . . . . . . . 10  |-  ( ph  ->  ( M `  ( `' M `  ( L `
 { s } ) ) )  =  ( L `  {
s } ) )
4112, 40eqtr4d 2319 . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { s } ) ) ) )
42 eqid 2284 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
431, 2, 9dvhlmod 30567 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
443, 42, 4lspsncl 15728 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  v  e.  V )  ->  ( N `  { v } )  e.  (
LSubSp `  U ) )
4543, 11, 44syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( N `  {
v } )  e.  ( LSubSp `  U )
)
461, 7, 2, 42, 9, 39mapdcnvcl 31109 . . . . . . . . . 10  |-  ( ph  ->  ( `' M `  ( L `  { s } ) )  e.  ( LSubSp `  U )
)
471, 2, 42, 7, 9, 45, 46mapd11 31096 . . . . . . . . 9  |-  ( ph  ->  ( ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { s } ) ) )  <->  ( N `  { v } )  =  ( `' M `  ( L `  {
s } ) ) ) )
4841, 47mpbid 203 . . . . . . . 8  |-  ( ph  ->  ( N `  {
v } )  =  ( `' M `  ( L `  { s } ) ) )
491, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18hdmaprnlem3N 31310 . . . . . . . 8  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
5048, 49eqnetrrd 2467 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( L `  { s } ) )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
5115, 35, 6lspsncl 15728 . . . . . . . . . . 11  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
5229, 34, 51syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
5352, 38eleqtrrd 2361 . . . . . . . . 9  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
541, 7, 9, 39, 53mapdcnv11N 31116 . . . . . . . 8  |-  ( ph  ->  ( ( `' M `  ( L `  {
s } ) )  =  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( L `  {
s } )  =  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
5554necon3bid 2482 . . . . . . 7  |-  ( ph  ->  ( ( `' M `  ( L `  {
s } ) )  =/=  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
5650, 55mpbid 203 . . . . . 6  |-  ( ph  ->  ( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) )
5756necomd 2530 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =/=  ( L `  {
s } ) )
5815, 16, 6, 28, 34, 32, 57lspdisj2 15874 . . . 4  |-  ( ph  ->  ( ( L `  { ( ( S `
 u )  .+b  s ) } )  i^i  ( L `  { s } ) )  =  { Q } )
5927, 58eleqtrd 2360 . . 3  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  { Q }
)
60 elsni 3665 . . 3  |-  ( ( s ( -g `  C
) ( S `  t ) )  e. 
{ Q }  ->  ( s ( -g `  C
) ( S `  t ) )  =  Q )
6159, 60syl 17 . 2  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  =  Q )
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19hdmaprnlem4tN 31312 . . . 4  |-  ( ph  ->  t  e.  V )
631, 2, 3, 5, 15, 8, 9, 62hdmapcl 31290 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
64 eqid 2284 . . . 4  |-  ( -g `  C )  =  (
-g `  C )
6515, 16, 64lmodsubeq0 15678 . . 3  |-  ( ( C  e.  LMod  /\  s  e.  D  /\  ( S `  t )  e.  D )  ->  (
( s ( -g `  C ) ( S `
 t ) )  =  Q  <->  s  =  ( S `  t ) ) )
6629, 32, 63, 65syl3anc 1184 . 2  |-  ( ph  ->  ( ( s (
-g `  C )
( S `  t
) )  =  Q  <-> 
s  =  ( S `
 t ) ) )
6761, 66mpbid 203 1  |-  ( ph  ->  s  =  ( S `
 t ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447    \ cdif 3150    i^i cin 3152   {csn 3641   `'ccnv 4687   ran crn 4689   ` cfv 5221  (class class class)co 5819   Basecbs 13142   +g cplusg 13202   0gc0g 13394   -gcsg 14359   LModclmod 15621   LSubSpclss 15683   LSpanclspn 15722   HLchlt 28807   LHypclh 29440   DVecHcdvh 30535  LCDualclcd 31043  mapdcmpd 31081  HDMapchdma 31250
This theorem is referenced by:  hdmaprnlem10N  31319
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-fal 1313  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-ot 3651  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-undef 6291  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-n0 9961  df-z 10020  df-uz 10226  df-fz 10777  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-sca 13218  df-vsca 13219  df-0g 13398  df-mre 13482  df-mrc 13483  df-acs 13485  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-mnd 14361  df-submnd 14410  df-grp 14483  df-minusg 14484  df-sbg 14485  df-subg 14612  df-cntz 14787  df-oppg 14813  df-lsm 14941  df-cmn 15085  df-abl 15086  df-mgp 15320  df-rng 15334  df-ur 15336  df-oppr 15399  df-dvdsr 15417  df-unit 15418  df-invr 15448  df-dvr 15459  df-drng 15508  df-lmod 15623  df-lss 15684  df-lsp 15723  df-lvec 15850  df-lsatoms 28433  df-lshyp 28434  df-lcv 28476  df-lfl 28515  df-lkr 28543  df-ldual 28581  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tgrp 30199  df-tendo 30211  df-edring 30213  df-dveca 30459  df-disoa 30486  df-dvech 30536  df-dib 30596  df-dic 30630  df-dih 30686  df-doch 30805  df-djh 30852  df-lcdual 31044  df-mapd 31082  df-hvmap 31214  df-hdmap1 31251  df-hdmap 31252
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