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Theorem hdmap14lem6 32575
Description: Case where  F is zero. (Contributed by NM, 1-Jun-2015.)
Hypotheses
Ref Expression
hdmap14lem1.h  |-  H  =  ( LHyp `  K
)
hdmap14lem1.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap14lem1.v  |-  V  =  ( Base `  U
)
hdmap14lem1.t  |-  .x.  =  ( .s `  U )
hdmap14lem3.o  |-  .0.  =  ( 0g `  U )
hdmap14lem1.r  |-  R  =  (Scalar `  U )
hdmap14lem1.b  |-  B  =  ( Base `  R
)
hdmap14lem1.z  |-  Z  =  ( 0g `  R
)
hdmap14lem1.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap14lem2.e  |-  .xb  =  ( .s `  C )
hdmap14lem1.l  |-  L  =  ( LSpan `  C )
hdmap14lem2.p  |-  P  =  (Scalar `  C )
hdmap14lem2.a  |-  A  =  ( Base `  P
)
hdmap14lem2.q  |-  Q  =  ( 0g `  P
)
hdmap14lem1.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmap14lem1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap14lem3.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap14lem6.f  |-  ( ph  ->  F  =  Z )
Assertion
Ref Expression
hdmap14lem6  |-  ( ph  ->  E! g  e.  A  ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
Distinct variable groups:    A, g    C, g    .xb , g    Q, g    S, g    g, X    ph, g
Allowed substitution hints:    B( g)    P( g)    R( g)    .x. ( g)    U( g)    F( g)    H( g)    K( g)    L( g)    V( g)    W( g)    .0. ( g)    Z( g)

Proof of Theorem hdmap14lem6
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 hdmap14lem1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 hdmap14lem1.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
3 hdmap14lem1.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3lcdlmod 32291 . . . . 5  |-  ( ph  ->  C  e.  LMod )
5 hdmap14lem2.p . . . . . 6  |-  P  =  (Scalar `  C )
6 hdmap14lem2.a . . . . . 6  |-  A  =  ( Base `  P
)
7 hdmap14lem2.q . . . . . 6  |-  Q  =  ( 0g `  P
)
85, 6, 7lmod0cl 15966 . . . . 5  |-  ( C  e.  LMod  ->  Q  e.  A )
94, 8syl 16 . . . 4  |-  ( ph  ->  Q  e.  A )
10 hdmap14lem1.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
11 hdmap14lem1.v . . . . . . 7  |-  V  =  ( Base `  U
)
12 eqid 2435 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
13 hdmap14lem1.s . . . . . . 7  |-  S  =  ( (HDMap `  K
) `  W )
14 hdmap14lem3.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
1514eldifad 3324 . . . . . . 7  |-  ( ph  ->  X  e.  V )
161, 10, 11, 2, 12, 13, 3, 15hdmapcl 32532 . . . . . 6  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
17 hdmap14lem2.e . . . . . . 7  |-  .xb  =  ( .s `  C )
18 eqid 2435 . . . . . . 7  |-  ( 0g
`  C )  =  ( 0g `  C
)
1912, 5, 17, 7, 18lmod0vs 15973 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  X )  e.  ( Base `  C
) )  ->  ( Q  .xb  ( S `  X ) )  =  ( 0g `  C
) )
204, 16, 19syl2anc 643 . . . . 5  |-  ( ph  ->  ( Q  .xb  ( S `  X )
)  =  ( 0g
`  C ) )
2120eqcomd 2440 . . . 4  |-  ( ph  ->  ( 0g `  C
)  =  ( Q 
.xb  ( S `  X ) ) )
22 oveq1 6080 . . . . . 6  |-  ( g  =  Q  ->  (
g  .xb  ( S `  X ) )  =  ( Q  .xb  ( S `  X )
) )
2322eqeq2d 2446 . . . . 5  |-  ( g  =  Q  ->  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  <->  ( 0g `  C )  =  ( Q  .xb  ( S `  X ) ) ) )
2423rspcev 3044 . . . 4  |-  ( ( Q  e.  A  /\  ( 0g `  C )  =  ( Q  .xb  ( S `  X ) ) )  ->  E. g  e.  A  ( 0g `  C )  =  ( g  .xb  ( S `  X ) ) )
259, 21, 24syl2anc 643 . . 3  |-  ( ph  ->  E. g  e.  A  ( 0g `  C )  =  ( g  .xb  ( S `  X ) ) )
26 hdmap14lem3.o . . . . . . . . . . 11  |-  .0.  =  ( 0g `  U )
271, 10, 11, 26, 2, 18, 12, 13, 3, 14hdmapnzcl 32547 . . . . . . . . . 10  |-  ( ph  ->  ( S `  X
)  e.  ( (
Base `  C )  \  { ( 0g `  C ) } ) )
28 eldifsni 3920 . . . . . . . . . 10  |-  ( ( S `  X )  e.  ( ( Base `  C )  \  {
( 0g `  C
) } )  -> 
( S `  X
)  =/=  ( 0g
`  C ) )
2927, 28syl 16 . . . . . . . . 9  |-  ( ph  ->  ( S `  X
)  =/=  ( 0g
`  C ) )
3029neneqd 2614 . . . . . . . 8  |-  ( ph  ->  -.  ( S `  X )  =  ( 0g `  C ) )
31303ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  ->  -.  ( S `  X
)  =  ( 0g
`  C ) )
32 simp3l 985 . . . . . . . . . . 11  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) ) )
3332eqcomd 2440 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( g  .xb  ( S `  X )
)  =  ( 0g
`  C ) )
341, 2, 3lcdlvec 32290 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  LVec )
35343ad2ant1 978 . . . . . . . . . . 11  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  ->  C  e.  LVec )
36 simp2l 983 . . . . . . . . . . 11  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
g  e.  A )
37163ad2ant1 978 . . . . . . . . . . 11  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( S `  X
)  e.  ( Base `  C ) )
3812, 17, 5, 6, 7, 18, 35, 36, 37lvecvs0or 16170 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( ( g  .xb  ( S `  X ) )  =  ( 0g
`  C )  <->  ( g  =  Q  \/  ( S `  X )  =  ( 0g `  C ) ) ) )
3933, 38mpbid 202 . . . . . . . . 9  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( g  =  Q  \/  ( S `  X )  =  ( 0g `  C ) ) )
4039orcomd 378 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( ( S `  X )  =  ( 0g `  C )  \/  g  =  Q ) )
4140ord 367 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( -.  ( S `
 X )  =  ( 0g `  C
)  ->  g  =  Q ) )
4231, 41mpd 15 . . . . . 6  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
g  =  Q )
43 simp3r 986 . . . . . . . . . . 11  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( 0g `  C
)  =  ( h 
.xb  ( S `  X ) ) )
4443eqcomd 2440 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( h  .xb  ( S `  X )
)  =  ( 0g
`  C ) )
45 simp2r 984 . . . . . . . . . . 11  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  ->  h  e.  A )
4612, 17, 5, 6, 7, 18, 35, 45, 37lvecvs0or 16170 . . . . . . . . . 10  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( ( h  .xb  ( S `  X ) )  =  ( 0g
`  C )  <->  ( h  =  Q  \/  ( S `  X )  =  ( 0g `  C ) ) ) )
4744, 46mpbid 202 . . . . . . . . 9  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( h  =  Q  \/  ( S `  X )  =  ( 0g `  C ) ) )
4847orcomd 378 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( ( S `  X )  =  ( 0g `  C )  \/  h  =  Q ) )
4948ord 367 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
( -.  ( S `
 X )  =  ( 0g `  C
)  ->  h  =  Q ) )
5031, 49mpd 15 . . . . . 6  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  ->  h  =  Q )
5142, 50eqtr4d 2470 . . . . 5  |-  ( (
ph  /\  ( g  e.  A  /\  h  e.  A )  /\  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )  -> 
g  =  h )
52513exp 1152 . . . 4  |-  ( ph  ->  ( ( g  e.  A  /\  h  e.  A )  ->  (
( ( 0g `  C )  =  ( g  .xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) )  ->  g  =  h ) ) )
5352ralrimivv 2789 . . 3  |-  ( ph  ->  A. g  e.  A  A. h  e.  A  ( ( ( 0g
`  C )  =  ( g  .xb  ( S `  X )
)  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) )  ->  g  =  h ) )
54 oveq1 6080 . . . . 5  |-  ( g  =  h  ->  (
g  .xb  ( S `  X ) )  =  ( h  .xb  ( S `  X )
) )
5554eqeq2d 2446 . . . 4  |-  ( g  =  h  ->  (
( 0g `  C
)  =  ( g 
.xb  ( S `  X ) )  <->  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) ) )
5655reu4 3120 . . 3  |-  ( E! g  e.  A  ( 0g `  C )  =  ( g  .xb  ( S `  X ) )  <->  ( E. g  e.  A  ( 0g `  C )  =  ( g  .xb  ( S `  X ) )  /\  A. g  e.  A  A. h  e.  A  (
( ( 0g `  C )  =  ( g  .xb  ( S `  X ) )  /\  ( 0g `  C )  =  ( h  .xb  ( S `  X ) ) )  ->  g  =  h ) ) )
5725, 53, 56sylanbrc 646 . 2  |-  ( ph  ->  E! g  e.  A  ( 0g `  C )  =  ( g  .xb  ( S `  X ) ) )
58 hdmap14lem6.f . . . . . . . 8  |-  ( ph  ->  F  =  Z )
5958oveq1d 6088 . . . . . . 7  |-  ( ph  ->  ( F  .x.  X
)  =  ( Z 
.x.  X ) )
601, 10, 3dvhlmod 31809 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
61 hdmap14lem1.r . . . . . . . . 9  |-  R  =  (Scalar `  U )
62 hdmap14lem1.t . . . . . . . . 9  |-  .x.  =  ( .s `  U )
63 hdmap14lem1.z . . . . . . . . 9  |-  Z  =  ( 0g `  R
)
6411, 61, 62, 63, 26lmod0vs 15973 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( Z  .x.  X )  =  .0.  )
6560, 15, 64syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( Z  .x.  X
)  =  .0.  )
6659, 65eqtrd 2467 . . . . . 6  |-  ( ph  ->  ( F  .x.  X
)  =  .0.  )
6766fveq2d 5724 . . . . 5  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( S `  .0.  ) )
681, 10, 26, 2, 18, 13, 3hdmapval0 32535 . . . . 5  |-  ( ph  ->  ( S `  .0.  )  =  ( 0g `  C ) )
6967, 68eqtrd 2467 . . . 4  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( 0g `  C ) )
7069eqeq1d 2443 . . 3  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  =  ( g 
.xb  ( S `  X ) )  <->  ( 0g `  C )  =  ( g  .xb  ( S `  X ) ) ) )
7170reubidv 2884 . 2  |-  ( ph  ->  ( E! g  e.  A  ( S `  ( F  .x.  X ) )  =  ( g 
.xb  ( S `  X ) )  <->  E! g  e.  A  ( 0g `  C )  =  ( g  .xb  ( S `  X ) ) ) )
7257, 71mpbird 224 1  |-  ( ph  ->  E! g  e.  A  ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   E!wreu 2699    \ cdif 3309   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13459  Scalarcsca 13522   .scvsca 13523   0gc0g 13713   LModclmod 15940   LSpanclspn 16037   LVecclvec 16164   HLchlt 30049   LHypclh 30682   DVecHcdvh 31777  LCDualclcd 32285  HDMapchdma 32492
This theorem is referenced by:  hdmap14lem7  32576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-0g 13717  df-mre 13801  df-mrc 13802  df-acs 13804  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-mnd 14680  df-submnd 14729  df-grp 14802  df-minusg 14803  df-sbg 14804  df-subg 14931  df-cntz 15106  df-oppg 15132  df-lsm 15260  df-cmn 15404  df-abl 15405  df-mgp 15639  df-rng 15653  df-ur 15655  df-oppr 15718  df-dvdsr 15736  df-unit 15737  df-invr 15767  df-dvr 15778  df-drng 15827  df-lmod 15942  df-lss 15999  df-lsp 16038  df-lvec 16165  df-lsatoms 29675  df-lshyp 29676  df-lcv 29718  df-lfl 29757  df-lkr 29785  df-ldual 29823  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197  df-lvols 30198  df-lines 30199  df-psubsp 30201  df-pmap 30202  df-padd 30494  df-lhyp 30686  df-laut 30687  df-ldil 30802  df-ltrn 30803  df-trl 30857  df-tgrp 31441  df-tendo 31453  df-edring 31455  df-dveca 31701  df-disoa 31728  df-dvech 31778  df-dib 31838  df-dic 31872  df-dih 31928  df-doch 32047  df-djh 32094  df-lcdual 32286  df-mapd 32324  df-hvmap 32456  df-hdmap1 32493  df-hdmap 32494
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