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Theorem hdmap14lem10 31221
Description: Part of proof of part 14 in [Baer] p. 49 line 38. (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 ) ) )
hdmap14lem8.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
Assertion
Ref Expression
hdmap14lem10  |-  ( ph  ->  G  =  I )

Proof of Theorem hdmap14lem10
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.t . . 3  |-  .x.  =  ( .s `  U )
5 hdmap14lem8.r . . 3  |-  R  =  (Scalar `  U )
6 hdmap14lem8.b . . 3  |-  B  =  ( Base `  R
)
7 hdmap14lem8.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap14lem8.e . . 3  |-  .xb  =  ( .s `  C )
9 eqid 2256 . . 3  |-  ( LSpan `  C )  =  (
LSpan `  C )
10 hdmap14lem8.p . . 3  |-  P  =  (Scalar `  C )
11 hdmap14lem8.a . . 3  |-  A  =  ( Base `  P
)
12 hdmap14lem8.s . . 3  |-  S  =  ( (HDMap `  K
) `  W )
13 hdmap14lem8.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
141, 2, 13dvhlmod 30451 . . . 4  |-  ( ph  ->  U  e.  LMod )
15 hdmap14lem8.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
16 eldifi 3259 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
1715, 16syl 17 . . . 4  |-  ( ph  ->  X  e.  V )
18 hdmap14lem8.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
19 eldifi 3259 . . . . 5  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
2018, 19syl 17 . . . 4  |-  ( ph  ->  Y  e.  V )
21 hdmap14lem8.q . . . . 5  |-  .+  =  ( +g  `  U )
223, 21lmodvacl 15589 . . . 4  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
2314, 17, 20, 22syl3anc 1187 . . 3  |-  ( ph  ->  ( X  .+  Y
)  e.  V )
24 hdmap14lem8.f . . 3  |-  ( ph  ->  F  e.  B )
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 23, 24hdmap14lem2a 31211 . 2  |-  ( ph  ->  E. g  e.  A  ( S `  ( F 
.x.  ( X  .+  Y ) ) )  =  ( g  .xb  ( S `  ( X 
.+  Y ) ) ) )
26 hdmap14lem8.o . . . 4  |-  .0.  =  ( 0g `  U )
27 hdmap14lem8.n . . . 4  |-  N  =  ( LSpan `  U )
28 hdmap14lem8.d . . . 4  |-  .+b  =  ( +g  `  C )
29133ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
30153ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  ->  X  e.  ( V  \  {  .0.  } ) )
31183ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  ->  Y  e.  ( V  \  {  .0.  } ) )
32243ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  ->  F  e.  B )
33 hdmap14lem8.g . . . . 5  |-  ( ph  ->  G  e.  A )
34333ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  ->  G  e.  A )
35 hdmap14lem8.i . . . . 5  |-  ( ph  ->  I  e.  A )
36353ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  ->  I  e.  A )
37 hdmap14lem8.xx . . . . 5  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )
38373ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  -> 
( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )
39 hdmap14lem8.yy . . . . 5  |-  ( ph  ->  ( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )
40393ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  -> 
( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )
41 hdmap14lem8.ne . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
42413ad2ant1 981 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  -> 
( N `  { X } )  =/=  ( N `  { Y } ) )
43 simp2 961 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  -> 
g  e.  A )
44 simp3 962 . . . 4  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  -> 
( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( g  .xb  ( S `  ( X 
.+  Y ) ) ) )
451, 2, 3, 21, 4, 26, 27, 5, 6, 7, 28, 8, 10, 11, 12, 29, 30, 31, 32, 34, 36, 38, 40, 42, 43, 44hdmap14lem9 31220 . . 3  |-  ( (
ph  /\  g  e.  A  /\  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) ) )  ->  G  =  I )
4645rexlimdv3a 2642 . 2  |-  ( ph  ->  ( E. g  e.  A  ( S `  ( F  .x.  ( X 
.+  Y ) ) )  =  ( g 
.xb  ( S `  ( X  .+  Y ) ) )  ->  G  =  I ) )
4725, 46mpd 16 1  |-  ( ph  ->  G  =  I )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517    \ cdif 3110   {csn 3600   ` cfv 4659  (class class class)co 5778   Basecbs 13096   +g cplusg 13156  Scalarcsca 13159   .scvsca 13160   0gc0g 13348   LModclmod 15575   LSpanclspn 15676   HLchlt 28691   LHypclh 29324   DVecHcdvh 30419  LCDualclcd 30927  HDMapchdma 31134
This theorem is referenced by:  hdmap14lem11  31222
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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