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Theorem hdmap14lem2a 35109
Description: Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include  F  =  .0. so it can be used in hdmap14lem10 35119. (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 6087 . . . . 5  |-  ( F  =  ( 0g `  R )  ->  ( F  .x.  X )  =  ( ( 0g `  R )  .x.  X
) )
21fveq2d 5683 . . . 4  |-  ( F  =  ( 0g `  R )  ->  ( S `  ( F  .x.  X ) )  =  ( S `  (
( 0g `  R
)  .x.  X )
) )
32eqeq1d 2441 . . 3  |-  ( F  =  ( 0g `  R )  ->  (
( S `  ( F  .x.  X ) )  =  ( g  .xb  ( S `  X ) )  <->  ( S `  ( ( 0g `  R )  .x.  X
) )  =  ( g  .xb  ( S `  X ) ) ) )
43rexbidv 2726 . 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 3471 . . 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 462 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
20 hdmap14lem3a.x . . . . . . 7  |-  ( ph  ->  X  e.  V )
2120adantr 462 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  X  e.  V )
22 hdmap14lem1a.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
2322adantr 462 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  F  e.  B )
24 eqid 2433 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
25 simpr 458 . . . . . 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 35108 . . . . 5  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( L `  { ( S `  X ) } )  =  ( L `  { ( S `  ( F  .x.  X ) ) } ) )
2726eqcomd 2438 . . . 4  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( L `  { ( S `  ( F  .x.  X ) ) } )  =  ( L `  {
( S `  X
) } ) )
28 eqid 2433 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
29 eqid 2433 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
306, 12, 18lcdlvec 34830 . . . . . 6  |-  ( ph  ->  C  e.  LVec )
316, 7, 18dvhlmod 34349 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
328, 10, 9, 11lmodvscl 16889 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  F  e.  B  /\  X  e.  V )  ->  ( F  .x.  X )  e.  V )
3331, 22, 20, 32syl3anc 1211 . . . . . . 7  |-  ( ph  ->  ( F  .x.  X
)  e.  V )
346, 7, 8, 12, 28, 17, 18, 33hdmapcl 35072 . . . . . 6  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  e.  ( Base `  C
) )
356, 7, 8, 12, 28, 17, 18, 20hdmapcl 35072 . . . . . 6  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
3628, 15, 16, 29, 13, 14, 30, 34, 35lspsneq 17125 . . . . 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 462 . . . 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 210 . . 3  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  E. g  e.  ( A  \  {
( 0g `  P
) } ) ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
39 ssrexv 3405 . . 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 63 . 2  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  E. g  e.  A  ( S `  ( F  .x.  X
) )  =  ( g  .xb  ( S `  X ) ) )
416, 12, 18lcdlmod 34831 . . . 4  |-  ( ph  ->  C  e.  LMod )
4215, 16, 29lmod0cl 16898 . . . 4  |-  ( C  e.  LMod  ->  ( 0g
`  P )  e.  A )
4341, 42syl 16 . . 3  |-  ( ph  ->  ( 0g `  P
)  e.  A )
44 eqid 2433 . . . . 5  |-  ( 0g
`  U )  =  ( 0g `  U
)
45 eqid 2433 . . . . 5  |-  ( 0g
`  C )  =  ( 0g `  C
)
466, 7, 44, 12, 45, 17, 18hdmapval0 35075 . . . 4  |-  ( ph  ->  ( S `  ( 0g `  U ) )  =  ( 0g `  C ) )
478, 10, 9, 24, 44lmod0vs 16905 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  R
)  .x.  X )  =  ( 0g `  U ) )
4831, 20, 47syl2anc 654 . . . . 5  |-  ( ph  ->  ( ( 0g `  R )  .x.  X
)  =  ( 0g
`  U ) )
4948fveq2d 5683 . . . 4  |-  ( ph  ->  ( S `  (
( 0g `  R
)  .x.  X )
)  =  ( S `
 ( 0g `  U ) ) )
5028, 15, 13, 29, 45lmod0vs 16905 . . . . 5  |-  ( ( C  e.  LMod  /\  ( S `  X )  e.  ( Base `  C
) )  ->  (
( 0g `  P
)  .xb  ( S `  X ) )  =  ( 0g `  C
) )
5141, 35, 50syl2anc 654 . . . 4  |-  ( ph  ->  ( ( 0g `  P )  .xb  ( S `  X )
)  =  ( 0g
`  C ) )
5246, 49, 513eqtr4d 2475 . . 3  |-  ( ph  ->  ( S `  (
( 0g `  R
)  .x.  X )
)  =  ( ( 0g `  P ) 
.xb  ( S `  X ) ) )
53 oveq1 6087 . . . . 5  |-  ( g  =  ( 0g `  P )  ->  (
g  .xb  ( S `  X ) )  =  ( ( 0g `  P )  .xb  ( S `  X )
) )
5453eqeq2d 2444 . . . 4  |-  ( g  =  ( 0g `  P )  ->  (
( S `  (
( 0g `  R
)  .x.  X )
)  =  ( g 
.xb  ( S `  X ) )  <->  ( S `  ( ( 0g `  R )  .x.  X
) )  =  ( ( 0g `  P
)  .xb  ( S `  X ) ) ) )
5554rspcev 3062 . . 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 654 . 2  |-  ( ph  ->  E. g  e.  A  ( S `  ( ( 0g `  R ) 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
574, 40, 56pm2.61ne 2676 1  |-  ( ph  ->  E. g  e.  A  ( S `  ( F 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706    \ cdif 3313    C_ wss 3316   {csn 3865   ` cfv 5406  (class class class)co 6080   Basecbs 14157  Scalarcsca 14224   .scvsca 14225   0gc0g 14361   LModclmod 16872   LSpanclspn 16974   HLchlt 32589   LHypclh 33222   DVecHcdvh 34317  LCDualclcd 34825  HDMapchdma 35032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-riotaBAD 32198
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-ot 3874  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-undef 6778  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-0g 14363  df-mre 14507  df-mrc 14508  df-acs 14510  df-poset 15099  df-plt 15111  df-lub 15127  df-glb 15128  df-join 15129  df-meet 15130  df-p0 15192  df-p1 15193  df-lat 15199  df-clat 15261  df-mnd 15398  df-submnd 15448  df-grp 15525  df-minusg 15526  df-sbg 15527  df-subg 15658  df-cntz 15815  df-oppg 15841  df-lsm 16115  df-cmn 16259  df-abl 16260  df-mgp 16566  df-rng 16580  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-unit 16668  df-invr 16698  df-dvr 16709  df-drng 16758  df-lmod 16874  df-lss 16936  df-lsp 16975  df-lvec 17106  df-lsatoms 32215  df-lshyp 32216  df-lcv 32258  df-lfl 32297  df-lkr 32325  df-ldual 32363  df-oposet 32415  df-ol 32417  df-oml 32418  df-covers 32505  df-ats 32506  df-atl 32537  df-cvlat 32561  df-hlat 32590  df-llines 32736  df-lplanes 32737  df-lvols 32738  df-lines 32739  df-psubsp 32741  df-pmap 32742  df-padd 33034  df-lhyp 33226  df-laut 33227  df-ldil 33342  df-ltrn 33343  df-trl 33397  df-tgrp 33981  df-tendo 33993  df-edring 33995  df-dveca 34241  df-disoa 34268  df-dvech 34318  df-dib 34378  df-dic 34412  df-dih 34468  df-doch 34587  df-djh 34634  df-lcdual 34826  df-mapd 34864  df-hvmap 34996  df-hdmap1 35033  df-hdmap 35034
This theorem is referenced by:  hdmap14lem10  35119  hdmap14lem11  35120  hdmap14lem12  35121
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