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Theorem hdmap14lem12 35525
Description: Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
Hypotheses
Ref Expression
hdmap14lem12.h  |-  H  =  ( LHyp `  K
)
hdmap14lem12.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap14lem12.v  |-  V  =  ( Base `  U
)
hdmap14lem12.t  |-  .x.  =  ( .s `  U )
hdmap14lem12.r  |-  R  =  (Scalar `  U )
hdmap14lem12.b  |-  B  =  ( Base `  R
)
hdmap14lem12.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap14lem12.e  |-  .xb  =  ( .s `  C )
hdmap14lem12.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmap14lem12.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap14lem12.f  |-  ( ph  ->  F  e.  B )
hdmap14lem12.p  |-  P  =  (Scalar `  C )
hdmap14lem12.a  |-  A  =  ( Base `  P
)
hdmap14lem12.o  |-  .0.  =  ( 0g `  U )
hdmap14lem12.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap14lem12.g  |-  ( ph  ->  G  e.  A )
Assertion
Ref Expression
hdmap14lem12  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) )  <->  A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
Distinct variable groups:    y, A    y, 
.xb    y, F    y, G    y,  .0.    y, S    y,  .x.    y, U    y, V    y, X    ph, y
Allowed substitution hints:    B( y)    C( y)    P( y)    R( y)    H( y)    K( y)    W( y)

Proof of Theorem hdmap14lem12
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 hdmap14lem12.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 hdmap14lem12.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap14lem12.v . . . . . 6  |-  V  =  ( Base `  U
)
4 hdmap14lem12.t . . . . . 6  |-  .x.  =  ( .s `  U )
5 hdmap14lem12.r . . . . . 6  |-  R  =  (Scalar `  U )
6 hdmap14lem12.b . . . . . 6  |-  B  =  ( Base `  R
)
7 hdmap14lem12.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap14lem12.e . . . . . 6  |-  .xb  =  ( .s `  C )
9 eqid 2442 . . . . . 6  |-  ( LSpan `  C )  =  (
LSpan `  C )
10 hdmap14lem12.p . . . . . 6  |-  P  =  (Scalar `  C )
11 hdmap14lem12.a . . . . . 6  |-  A  =  ( Base `  P
)
12 hdmap14lem12.s . . . . . 6  |-  S  =  ( (HDMap `  K
) `  W )
13 hdmap14lem12.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
14133ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 simp3 990 . . . . . . 7  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  y  e.  ( V  \  {  .0.  } ) )
1615eldifad 3339 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  y  e.  V )
17 hdmap14lem12.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
18173ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  F  e.  B )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18hdmap14lem2a 35513 . . . . 5  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  E. g  e.  A  ( S `  ( F  .x.  y
) )  =  ( g  .xb  ( S `  y ) ) )
20 simp3 990 . . . . . . 7  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )
21 eqid 2442 . . . . . . . . 9  |-  ( +g  `  U )  =  ( +g  `  U )
22 hdmap14lem12.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
23 eqid 2442 . . . . . . . . 9  |-  ( LSpan `  U )  =  (
LSpan `  U )
24 eqid 2442 . . . . . . . . 9  |-  ( +g  `  C )  =  ( +g  `  C )
25 simp11 1018 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  ph )
2625, 13syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
27 hdmap14lem12.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2825, 27syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  X  e.  ( V  \  {  .0.  } ) )
29 simp13 1020 . . . . . . . . 9  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  y  e.  ( V  \  {  .0.  } ) )
3025, 17syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  F  e.  B )
31 hdmap14lem12.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  A )
3225, 31syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  G  e.  A )
33 simp2 989 . . . . . . . . 9  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  g  e.  A )
34 simp12 1019 . . . . . . . . 9  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
) )
351, 2, 3, 21, 4, 22, 23, 5, 6, 7, 24, 8, 10, 11, 12, 26, 28, 29, 30, 32, 33, 34, 20hdmap14lem11 35524 . . . . . . . 8  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  G  =  g )
3635oveq1d 6105 . . . . . . 7  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  ( G  .xb  ( S `  y ) )  =  ( g  .xb  ( S `  y )
) )
3720, 36eqtr4d 2477 . . . . . 6  |-  ( ( ( ph  /\  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
)  /\  y  e.  ( V  \  {  .0.  } ) )  /\  g  e.  A  /\  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
) )  ->  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) )
3837rexlimdv3a 2842 . . . . 5  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( E. g  e.  A  ( S `  ( F  .x.  y ) )  =  ( g  .xb  ( S `  y )
)  ->  ( S `  ( F  .x.  y
) )  =  ( G  .xb  ( S `  y ) ) ) )
3919, 38mpd 15 . . . 4  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( S `  ( F  .x.  y
) )  =  ( G  .xb  ( S `  y ) ) )
40393expia 1189 . . 3  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) )  ->  ( y  e.  ( V  \  {  .0.  } )  ->  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
4140ralrimiv 2797 . 2  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) )  ->  A. y  e.  ( V  \  {  .0.  } ) ( S `  ( F  .x.  y ) )  =  ( G 
.xb  ( S `  y ) ) )
42 oveq2 6098 . . . . . . 7  |-  ( y  =  X  ->  ( F  .x.  y )  =  ( F  .x.  X
) )
4342fveq2d 5694 . . . . . 6  |-  ( y  =  X  ->  ( S `  ( F  .x.  y ) )  =  ( S `  ( F  .x.  X ) ) )
44 fveq2 5690 . . . . . . 7  |-  ( y  =  X  ->  ( S `  y )  =  ( S `  X ) )
4544oveq2d 6106 . . . . . 6  |-  ( y  =  X  ->  ( G  .xb  ( S `  y ) )  =  ( G  .xb  ( S `  X )
) )
4643, 45eqeq12d 2456 . . . . 5  |-  ( y  =  X  ->  (
( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  <->  ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) ) ) )
4746rspcv 3068 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  -> 
( A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
)  ->  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) ) )
4827, 47syl 16 . . 3  |-  ( ph  ->  ( A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
)  ->  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) ) )
4948imp 429 . 2  |-  ( (
ph  /\  A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) )  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
) )
5041, 49impbida 828 1  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) )  <->  A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   E.wrex 2715    \ cdif 3324   {csn 3876   ` cfv 5417  (class class class)co 6090   Basecbs 14173   +g cplusg 14237  Scalarcsca 14240   .scvsca 14241   0gc0g 14377   LSpanclspn 17051   HLchlt 32993   LHypclh 33626   DVecHcdvh 34721  LCDualclcd 35229  HDMapchdma 35436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-riotaBAD 32602
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-ot 3885  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6791  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-0g 14379  df-mre 14523  df-mrc 14524  df-acs 14526  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-subg 15677  df-cntz 15834  df-oppg 15860  df-lsm 16134  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-drng 16833  df-lmod 16949  df-lss 17013  df-lsp 17052  df-lvec 17183  df-lsatoms 32619  df-lshyp 32620  df-lcv 32662  df-lfl 32701  df-lkr 32729  df-ldual 32767  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-llines 33140  df-lplanes 33141  df-lvols 33142  df-lines 33143  df-psubsp 33145  df-pmap 33146  df-padd 33438  df-lhyp 33630  df-laut 33631  df-ldil 33746  df-ltrn 33747  df-trl 33801  df-tgrp 34385  df-tendo 34397  df-edring 34399  df-dveca 34645  df-disoa 34672  df-dvech 34722  df-dib 34782  df-dic 34816  df-dih 34872  df-doch 34991  df-djh 35038  df-lcdual 35230  df-mapd 35268  df-hvmap 35400  df-hdmap1 35437  df-hdmap 35438
This theorem is referenced by:  hdmap14lem13  35526
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