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Theorem hdmap14lem2a 37337
Description: Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include  F  =  .0. so it can be used in hdmap14lem10 37347. (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 6288 . . . . 5  |-  ( F  =  ( 0g `  R )  ->  ( F  .x.  X )  =  ( ( 0g `  R )  .x.  X
) )
21fveq2d 5860 . . . 4  |-  ( F  =  ( 0g `  R )  ->  ( S `  ( F  .x.  X ) )  =  ( S `  (
( 0g `  R
)  .x.  X )
) )
32eqeq1d 2445 . . 3  |-  ( F  =  ( 0g `  R )  ->  (
( S `  ( F  .x.  X ) )  =  ( g  .xb  ( S `  X ) )  <->  ( S `  ( ( 0g `  R )  .x.  X
) )  =  ( g  .xb  ( S `  X ) ) ) )
43rexbidv 2954 . 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 3616 . . 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 465 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
20 hdmap14lem3a.x . . . . . . 7  |-  ( ph  ->  X  e.  V )
2120adantr 465 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  X  e.  V )
22 hdmap14lem1a.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
2322adantr 465 . . . . . 6  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  F  e.  B )
24 eqid 2443 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
25 simpr 461 . . . . . 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 37336 . . . . 5  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( L `  { ( S `  X ) } )  =  ( L `  { ( S `  ( F  .x.  X ) ) } ) )
2726eqcomd 2451 . . . 4  |-  ( (
ph  /\  F  =/=  ( 0g `  R ) )  ->  ( L `  { ( S `  ( F  .x.  X ) ) } )  =  ( L `  {
( S `  X
) } ) )
28 eqid 2443 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
29 eqid 2443 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
306, 12, 18lcdlvec 37058 . . . . . 6  |-  ( ph  ->  C  e.  LVec )
316, 7, 18dvhlmod 36577 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
328, 10, 9, 11lmodvscl 17403 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  F  e.  B  /\  X  e.  V )  ->  ( F  .x.  X )  e.  V )
3331, 22, 20, 32syl3anc 1229 . . . . . . 7  |-  ( ph  ->  ( F  .x.  X
)  e.  V )
346, 7, 8, 12, 28, 17, 18, 33hdmapcl 37300 . . . . . 6  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  e.  ( Base `  C
) )
356, 7, 8, 12, 28, 17, 18, 20hdmapcl 37300 . . . . . 6  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
3628, 15, 16, 29, 13, 14, 30, 34, 35lspsneq 17642 . . . . 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 465 . . . 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 3550 . . 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 37059 . . . 4  |-  ( ph  ->  C  e.  LMod )
4215, 16, 29lmod0cl 17412 . . . 4  |-  ( C  e.  LMod  ->  ( 0g
`  P )  e.  A )
4341, 42syl 16 . . 3  |-  ( ph  ->  ( 0g `  P
)  e.  A )
44 eqid 2443 . . . . 5  |-  ( 0g
`  U )  =  ( 0g `  U
)
45 eqid 2443 . . . . 5  |-  ( 0g
`  C )  =  ( 0g `  C
)
466, 7, 44, 12, 45, 17, 18hdmapval0 37303 . . . 4  |-  ( ph  ->  ( S `  ( 0g `  U ) )  =  ( 0g `  C ) )
478, 10, 9, 24, 44lmod0vs 17419 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  R
)  .x.  X )  =  ( 0g `  U ) )
4831, 20, 47syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( 0g `  R )  .x.  X
)  =  ( 0g
`  U ) )
4948fveq2d 5860 . . . 4  |-  ( ph  ->  ( S `  (
( 0g `  R
)  .x.  X )
)  =  ( S `
 ( 0g `  U ) ) )
5028, 15, 13, 29, 45lmod0vs 17419 . . . . 5  |-  ( ( C  e.  LMod  /\  ( S `  X )  e.  ( Base `  C
) )  ->  (
( 0g `  P
)  .xb  ( S `  X ) )  =  ( 0g `  C
) )
5141, 35, 50syl2anc 661 . . . 4  |-  ( ph  ->  ( ( 0g `  P )  .xb  ( S `  X )
)  =  ( 0g
`  C ) )
5246, 49, 513eqtr4d 2494 . . 3  |-  ( ph  ->  ( S `  (
( 0g `  R
)  .x.  X )
)  =  ( ( 0g `  P ) 
.xb  ( S `  X ) ) )
53 oveq1 6288 . . . . 5  |-  ( g  =  ( 0g `  P )  ->  (
g  .xb  ( S `  X ) )  =  ( ( 0g `  P )  .xb  ( S `  X )
) )
5453eqeq2d 2457 . . . 4  |-  ( g  =  ( 0g `  P )  ->  (
( S `  (
( 0g `  R
)  .x.  X )
)  =  ( g 
.xb  ( S `  X ) )  <->  ( S `  ( ( 0g `  R )  .x.  X
) )  =  ( ( 0g `  P
)  .xb  ( S `  X ) ) ) )
5554rspcev 3196 . . 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 661 . 2  |-  ( ph  ->  E. g  e.  A  ( S `  ( ( 0g `  R ) 
.x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
574, 40, 56pm2.61ne 2758 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 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794    \ cdif 3458    C_ wss 3461   {csn 4014   ` cfv 5578  (class class class)co 6281   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   0gc0g 14714   LModclmod 17386   LSpanclspn 17491   HLchlt 34815   LHypclh 35448   DVecHcdvh 36545  LCDualclcd 37053  HDMapchdma 37260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-riotaBAD 34424
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-undef 7004  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-0g 14716  df-mre 14860  df-mrc 14861  df-acs 14863  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-subg 16072  df-cntz 16229  df-oppg 16255  df-lsm 16530  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-dvr 17206  df-drng 17272  df-lmod 17388  df-lss 17453  df-lsp 17492  df-lvec 17623  df-lsatoms 34441  df-lshyp 34442  df-lcv 34484  df-lfl 34523  df-lkr 34551  df-ldual 34589  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-llines 34962  df-lplanes 34963  df-lvols 34964  df-lines 34965  df-psubsp 34967  df-pmap 34968  df-padd 35260  df-lhyp 35452  df-laut 35453  df-ldil 35568  df-ltrn 35569  df-trl 35624  df-tgrp 36209  df-tendo 36221  df-edring 36223  df-dveca 36469  df-disoa 36496  df-dvech 36546  df-dib 36606  df-dic 36640  df-dih 36696  df-doch 36815  df-djh 36862  df-lcdual 37054  df-mapd 37092  df-hvmap 37224  df-hdmap1 37261  df-hdmap 37262
This theorem is referenced by:  hdmap14lem10  37347  hdmap14lem11  37348  hdmap14lem12  37349
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