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Theorem hdmap14lem2a 32507
Description: Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include  F  =  .0. so it can be used in hdmap14lem10 32517. (Contributed by NM, 31-May-2015.)
Hypotheses
Ref Expression
hdmap14lem1a.h  |-  H  =  ( LHyp `  K
)
hdmap14lem1a.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap14lem1a.v  |-  V  =  ( Base `  U
)
hdmap14lem1a.t  |-  .x.  =  ( .s `  U )
hdmap14lem1a.r  |-  R  =  (Scalar `  U )
hdmap14lem1a.b  |-  B  =  ( Base `  R
)
hdmap14lem1a.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap14lem2a.e  |-  .xb  =  ( .s `  C )
hdmap14lem1a.l  |-  L  =  ( LSpan `  C )
hdmap14lem2a.p  |-  P  =  (Scalar `  C )
hdmap14lem2a.a  |-  A  =  ( Base `  P
)
hdmap14lem1a.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmap14lem1a.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap14lem3a.x  |-  ( ph  ->  X  e.  V )
hdmap14lem1a.f  |-  ( ph  ->  F  e.  B )
Assertion
Ref Expression
hdmap14lem2a  |-  ( ph  ->  E. g  e.  A  ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
Distinct variable groups:    A, g    g, F    P, g    R, g    .x. , g    .xb , g    S, g    g, X
Allowed substitution hints:    ph( g)    B( g)    C( g)    U( g)    H( g)    K( g)    L( g)    V( g)    W( g)

Proof of Theorem hdmap14lem2a
StepHypRef Expression
1 oveq1 6079 . . . . 5  |-  ( F  =  ( 0g `  R )  ->  ( F  .x.  X )  =  ( ( 0g `  R )  .x.  X
) )
21fveq2d 5723 . . . 4  |-  ( F  =  ( 0g `  R )  ->  ( S `  ( F  .x.  X ) )  =  ( S `  (
( 0g `  R
)  .x.  X )
) )
32eqeq1d 2443 . . 3  |-  ( F  =  ( 0g `  R )  ->  (
( S `  ( F  .x.  X ) )  =  ( g  .xb  ( S `  X ) )  <->  ( S `  ( ( 0g `  R )  .x.  X
) )  =  ( g  .xb  ( S `  X ) ) ) )
43rexbidv 2718 . 2  |-  ( F  =  ( 0g `  R )  ->  ( E. g  e.  A  ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) )  <->  E. g  e.  A  ( S `  ( ( 0g `  R ) 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) ) )
5 difss 3466 . . 3  |-  ( A 
\  { ( 0g
`  P ) } )  C_  A
6 hdmap14lem1a.h . . . . . 6  |-  H  =  ( LHyp `  K
)
7 hdmap14lem1a.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
8 hdmap14lem1a.v . . . . . 6  |-  V  =  ( Base `  U
)
9 hdmap14lem1a.t . . . . . 6  |-  .x.  =  ( .s `  U )
10 hdmap14lem1a.r . . . . . 6  |-  R  =  (Scalar `  U )
11 hdmap14lem1a.b . . . . . 6  |-  B  =  ( Base `  R
)
12 hdmap14lem1a.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
13 hdmap14lem2a.e . . . . . 6  |-  .xb  =  ( .s `  C )
14 hdmap14lem1a.l . . . . . 6  |-  L  =  ( LSpan `  C )
15 hdmap14lem2a.p . . . . . 6  |-  P  =  (Scalar `  C )
16 hdmap14lem2a.a . . . . . 6  |-  A  =  ( Base `  P
)
17 hdmap14lem1a.s . . . . . 6  |-  S  =  ( (HDMap `  K
) `  W )
18 hdmap14lem1a.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
1918adantr 452 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
20 hdmap14lem3a.x . . . . . . 7  |-  ( ph  ->  X  e.  V )
2120adantr 452 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  X  e.  V )
22 hdmap14lem1a.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
2322adantr 452 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  F  e.  B )
24 eqid 2435 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
25 simpr 448 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  F  =/=  ( 0g `  R ) )
266, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23, 24, 25hdmap14lem1a 32506 . . . . 5  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( L `  { ( S `  X ) } )  =  ( L `  { ( S `  ( F  .x.  X ) ) } ) )
2726eqcomd 2440 . . . 4  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( L `  { ( S `  ( F  .x.  X ) ) } )  =  ( L `  {
( S `  X
) } ) )
28 eqid 2435 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
29 eqid 2435 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
306, 12, 18lcdlvec 32228 . . . . . 6  |-  ( ph  ->  C  e.  LVec )
316, 7, 18dvhlmod 31747 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
328, 10, 9, 11lmodvscl 15955 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  F  e.  B  /\  X  e.  V )  ->  ( F  .x.  X )  e.  V )
3331, 22, 20, 32syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( F  .x.  X
)  e.  V )
346, 7, 8, 12, 28, 17, 18, 33hdmapcl 32470 . . . . . 6  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  e.  ( Base `  C
) )
356, 7, 8, 12, 28, 17, 18, 20hdmapcl 32470 . . . . . 6  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
3628, 15, 16, 29, 13, 14, 30, 34, 35lspsneq 16182 . . . . 5  |-  ( ph  ->  ( ( L `  { ( S `  ( F  .x.  X ) ) } )  =  ( L `  {
( S `  X
) } )  <->  E. g  e.  ( A  \  {
( 0g `  P
) } ) ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) ) )
3736adantr 452 . . . 4  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( ( L `  { ( S `  ( F  .x.  X ) ) } )  =  ( L `
 { ( S `
 X ) } )  <->  E. g  e.  ( A  \  { ( 0g `  P ) } ) ( S `
 ( F  .x.  X ) )  =  ( g  .xb  ( S `  X )
) ) )
3827, 37mpbid 202 . . 3  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  E. g  e.  ( A  \  {
( 0g `  P
) } ) ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
39 ssrexv 3400 . . 3  |-  ( ( A  \  { ( 0g `  P ) } )  C_  A  ->  ( E. g  e.  ( A  \  {
( 0g `  P
) } ) ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) )  ->  E. g  e.  A  ( S `  ( F  .x.  X
) )  =  ( g  .xb  ( S `  X ) ) ) )
405, 38, 39mpsyl 61 . 2  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  E. g  e.  A  ( S `  ( F  .x.  X
) )  =  ( g  .xb  ( S `  X ) ) )
416, 12, 18lcdlmod 32229 . . . 4  |-  ( ph  ->  C  e.  LMod )
4215, 16, 29lmod0cl 15964 . . . 4  |-  ( C  e.  LMod  ->  ( 0g
`  P )  e.  A )
4341, 42syl 16 . . 3  |-  ( ph  ->  ( 0g `  P
)  e.  A )
44 eqid 2435 . . . . 5  |-  ( 0g
`  U )  =  ( 0g `  U
)
45 eqid 2435 . . . . 5  |-  ( 0g
`  C )  =  ( 0g `  C
)
466, 7, 44, 12, 45, 17, 18hdmapval0 32473 . . . 4  |-  ( ph  ->  ( S `  ( 0g `  U ) )  =  ( 0g `  C ) )
478, 10, 9, 24, 44lmod0vs 15971 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  R
)  .x.  X )  =  ( 0g `  U ) )
4831, 20, 47syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( 0g `  R )  .x.  X
)  =  ( 0g
`  U ) )
4948fveq2d 5723 . . . 4  |-  ( ph  ->  ( S `  (
( 0g `  R
)  .x.  X )
)  =  ( S `
 ( 0g `  U ) ) )
5028, 15, 13, 29, 45lmod0vs 15971 . . . . 5  |-  ( ( C  e.  LMod  /\  ( S `  X )  e.  ( Base `  C
) )  ->  (
( 0g `  P
)  .xb  ( S `  X ) )  =  ( 0g `  C
) )
5141, 35, 50syl2anc 643 . . . 4  |-  ( ph  ->  ( ( 0g `  P )  .xb  ( S `  X )
)  =  ( 0g
`  C ) )
5246, 49, 513eqtr4d 2477 . . 3  |-  ( ph  ->  ( S `  (
( 0g `  R
)  .x.  X )
)  =  ( ( 0g `  P ) 
.xb  ( S `  X ) ) )
53 oveq1 6079 . . . . 5  |-  ( g  =  ( 0g `  P )  ->  (
g  .xb  ( S `  X ) )  =  ( ( 0g `  P )  .xb  ( S `  X )
) )
5453eqeq2d 2446 . . . 4  |-  ( g  =  ( 0g `  P )  ->  (
( S `  (
( 0g `  R
)  .x.  X )
)  =  ( g 
.xb  ( S `  X ) )  <->  ( S `  ( ( 0g `  R )  .x.  X
) )  =  ( ( 0g `  P
)  .xb  ( S `  X ) ) ) )
5554rspcev 3044 . . 3  |-  ( ( ( 0g `  P
)  e.  A  /\  ( S `  ( ( 0g `  R ) 
.x.  X ) )  =  ( ( 0g
`  P )  .xb  ( S `  X ) ) )  ->  E. g  e.  A  ( S `  ( ( 0g `  R )  .x.  X
) )  =  ( g  .xb  ( S `  X ) ) )
5643, 52, 55syl2anc 643 . 2  |-  ( ph  ->  E. g  e.  A  ( S `  ( ( 0g `  R ) 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
574, 40, 56pm2.61ne 2673 1  |-  ( ph  ->  E. g  e.  A  ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309    C_ wss 3312   {csn 3806   ` cfv 5445  (class class class)co 6072   Basecbs 13457  Scalarcsca 13520   .scvsca 13521   0gc0g 13711   LModclmod 15938   LSpanclspn 16035   HLchlt 29987   LHypclh 30620   DVecHcdvh 31715  LCDualclcd 32223  HDMapchdma 32430
This theorem is referenced by:  hdmap14lem10  32517  hdmap14lem11  32518  hdmap14lem12  32519
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 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
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 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-tpos 6470  df-undef 6534  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-0g 13715  df-mre 13799  df-mrc 13800  df-acs 13802  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-subg 14929  df-cntz 15104  df-oppg 15130  df-lsm 15258  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-ur 15653  df-oppr 15716  df-dvdsr 15734  df-unit 15735  df-invr 15765  df-dvr 15776  df-drng 15825  df-lmod 15940  df-lss 15997  df-lsp 16036  df-lvec 16163  df-lsatoms 29613  df-lshyp 29614  df-lcv 29656  df-lfl 29695  df-lkr 29723  df-ldual 29761  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tgrp 31379  df-tendo 31391  df-edring 31393  df-dveca 31639  df-disoa 31666  df-dvech 31716  df-dib 31776  df-dic 31810  df-dih 31866  df-doch 31985  df-djh 32032  df-lcdual 32224  df-mapd 32262  df-hvmap 32394  df-hdmap1 32431  df-hdmap 32432
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