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Theorem hdmap14lem11 32144
Description: Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 3-Jun-2015.)
Hypotheses
Ref Expression
hdmap14lem8.h  |-  H  =  ( LHyp `  K
)
hdmap14lem8.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap14lem8.v  |-  V  =  ( Base `  U
)
hdmap14lem8.q  |-  .+  =  ( +g  `  U )
hdmap14lem8.t  |-  .x.  =  ( .s `  U )
hdmap14lem8.o  |-  .0.  =  ( 0g `  U )
hdmap14lem8.n  |-  N  =  ( LSpan `  U )
hdmap14lem8.r  |-  R  =  (Scalar `  U )
hdmap14lem8.b  |-  B  =  ( Base `  R
)
hdmap14lem8.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap14lem8.d  |-  .+b  =  ( +g  `  C )
hdmap14lem8.e  |-  .xb  =  ( .s `  C )
hdmap14lem8.p  |-  P  =  (Scalar `  C )
hdmap14lem8.a  |-  A  =  ( Base `  P
)
hdmap14lem8.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmap14lem8.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap14lem8.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap14lem8.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
hdmap14lem8.f  |-  ( ph  ->  F  e.  B )
hdmap14lem8.g  |-  ( ph  ->  G  e.  A )
hdmap14lem8.i  |-  ( ph  ->  I  e.  A )
hdmap14lem8.xx  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )
hdmap14lem8.yy  |-  ( ph  ->  ( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )
Assertion
Ref Expression
hdmap14lem11  |-  ( ph  ->  G  =  I )

Proof of Theorem hdmap14lem11
Dummy variables  g 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap14lem8.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap14lem8.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap14lem8.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap14lem8.n . . 3  |-  N  =  ( LSpan `  U )
5 hdmap14lem8.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
6 hdmap14lem8.x . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
7 eldifi 3300 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
86, 7syl 15 . . 3  |-  ( ph  ->  X  e.  V )
9 hdmap14lem8.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
10 eldifi 3300 . . . 4  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
119, 10syl 15 . . 3  |-  ( ph  ->  Y  e.  V )
121, 2, 3, 4, 5, 8, 11dvh3dim 31709 . 2  |-  ( ph  ->  E. z  e.  V  -.  z  e.  ( N `  { X ,  Y } ) )
13 hdmap14lem8.t . . . . 5  |-  .x.  =  ( .s `  U )
14 hdmap14lem8.r . . . . 5  |-  R  =  (Scalar `  U )
15 hdmap14lem8.b . . . . 5  |-  B  =  ( Base `  R
)
16 hdmap14lem8.c . . . . 5  |-  C  =  ( (LCDual `  K
) `  W )
17 hdmap14lem8.e . . . . 5  |-  .xb  =  ( .s `  C )
18 eqid 2285 . . . . 5  |-  ( LSpan `  C )  =  (
LSpan `  C )
19 hdmap14lem8.p . . . . 5  |-  P  =  (Scalar `  C )
20 hdmap14lem8.a . . . . 5  |-  A  =  ( Base `  P
)
21 hdmap14lem8.s . . . . 5  |-  S  =  ( (HDMap `  K
) `  W )
2253ad2ant1 976 . . . . 5  |-  ( (
ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
23 simp2 956 . . . . 5  |-  ( (
ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  ->  z  e.  V )
24 hdmap14lem8.f . . . . . 6  |-  ( ph  ->  F  e.  B )
25243ad2ant1 976 . . . . 5  |-  ( (
ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  ->  F  e.  B )
261, 2, 3, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25hdmap14lem2a 32133 . . . 4  |-  ( (
ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  ->  E. g  e.  A  ( S `  ( F  .x.  z
) )  =  ( g  .xb  ( S `  z ) ) )
27 hdmap14lem8.q . . . . . . 7  |-  .+  =  ( +g  `  U )
28 hdmap14lem8.o . . . . . . 7  |-  .0.  =  ( 0g `  U )
29 hdmap14lem8.d . . . . . . 7  |-  .+b  =  ( +g  `  C )
30 simp11 985 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ph )
3130, 5syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
32 eqid 2285 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
331, 2, 5dvhlmod 31373 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
3430, 33syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  U  e.  LMod )
353, 32, 4, 33, 8, 11lspprcl 15737 . . . . . . . . 9  |-  ( ph  ->  ( N `  { X ,  Y }
)  e.  ( LSubSp `  U ) )
3630, 35syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ( N `  { X ,  Y } )  e.  ( LSubSp `  U )
)
37 simp12 986 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  z  e.  V )
38 simp13 987 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  -.  z  e.  ( N `  { X ,  Y } ) )
393, 28, 32, 34, 36, 37, 38lssneln0 15711 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  z  e.  ( V  \  {  .0.  } ) )
4030, 6syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  X  e.  ( V  \  {  .0.  } ) )
4130, 24syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  F  e.  B )
42 simp2 956 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  g  e.  A )
43 hdmap14lem8.g . . . . . . . 8  |-  ( ph  ->  G  e.  A )
4430, 43syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  G  e.  A )
45 simp3 957 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )
46 hdmap14lem8.xx . . . . . . . 8  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )
4730, 46syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
) )
481, 2, 5dvhlvec 31372 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LVec )
4930, 48syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  U  e.  LVec )
5030, 8syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  X  e.  V )
5130, 11syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  Y  e.  V )
523, 4, 49, 37, 50, 51, 38lspindpi 15887 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  (
( N `  {
z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { Y } ) ) )
5352simpld 445 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ( N `  { z } )  =/=  ( N `  { X } ) )
541, 2, 3, 27, 13, 28, 4, 14, 15, 16, 29, 17, 19, 20, 21, 31, 39, 40, 41, 42, 44, 45, 47, 53hdmap14lem10 32143 . . . . . 6  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  g  =  G )
5530, 9syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  Y  e.  ( V  \  {  .0.  } ) )
56 hdmap14lem8.i . . . . . . . 8  |-  ( ph  ->  I  e.  A )
5730, 56syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  I  e.  A )
58 hdmap14lem8.yy . . . . . . . 8  |-  ( ph  ->  ( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )
5930, 58syl 15 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y )
) )
6052simprd 449 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  ( N `  { z } )  =/=  ( N `  { Y } ) )
611, 2, 3, 27, 13, 28, 4, 14, 15, 16, 29, 17, 19, 20, 21, 31, 39, 55, 41, 42, 57, 45, 59, 60hdmap14lem10 32143 . . . . . 6  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  g  =  I )
6254, 61eqtr3d 2319 . . . . 5  |-  ( ( ( ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  /\  g  e.  A  /\  ( S `  ( F  .x.  z ) )  =  ( g  .xb  ( S `  z )
) )  ->  G  =  I )
6362rexlimdv3a 2671 . . . 4  |-  ( (
ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  ->  ( E. g  e.  A  ( S `  ( F 
.x.  z ) )  =  ( g  .xb  ( S `  z ) )  ->  G  =  I ) )
6426, 63mpd 14 . . 3  |-  ( (
ph  /\  z  e.  V  /\  -.  z  e.  ( N `  { X ,  Y }
) )  ->  G  =  I )
6564rexlimdv3a 2671 . 2  |-  ( ph  ->  ( E. z  e.  V  -.  z  e.  ( N `  { X ,  Y }
)  ->  G  =  I ) )
6612, 65mpd 14 1  |-  ( ph  ->  G  =  I )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   E.wrex 2546    \ cdif 3151   {csn 3642   {cpr 3643   ` cfv 5257  (class class class)co 5860   Basecbs 13150   +g cplusg 13210  Scalarcsca 13213   .scvsca 13214   0gc0g 13402   LModclmod 15629   LSubSpclss 15691   LSpanclspn 15730   LVecclvec 15857   HLchlt 29613   LHypclh 30246   DVecHcdvh 31341  LCDualclcd 31849  HDMapchdma 32056
This theorem is referenced by:  hdmap14lem12  32145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-ot 3652  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-mre 13490  df-mrc 13491  df-acs 13493  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-cntz 14795  df-oppg 14821  df-lsm 14949  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-lsatoms 29239  df-lshyp 29240  df-lcv 29282  df-lfl 29321  df-lkr 29349  df-ldual 29387  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tgrp 31005  df-tendo 31017  df-edring 31019  df-dveca 31265  df-disoa 31292  df-dvech 31342  df-dib 31402  df-dic 31436  df-dih 31492  df-doch 31611  df-djh 31658  df-lcdual 31850  df-mapd 31888  df-hvmap 32020  df-hdmap1 32057  df-hdmap 32058
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