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Theorem hdmap14lem12 30976
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
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 2253 . . . . . 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 981 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 simp3 962 . . . . . . 7  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  y  e.  ( V  \  {  .0.  } ) )
16 eldifi 3215 . . . . . . 7  |-  ( y  e.  ( V  \  {  .0.  } )  -> 
y  e.  V )
1715, 16syl 17 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  y  e.  V )
18 hdmap14lem12.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
19183ad2ant1 981 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  F  e.  B )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 19hdmap14lem2a 30964 . . . . 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 ) ) )
21 simp3 962 . . . . . . 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 )
) )
22 eqid 2253 . . . . . . . . 9  |-  ( +g  `  U )  =  ( +g  `  U )
23 hdmap14lem12.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
24 eqid 2253 . . . . . . . . 9  |-  ( LSpan `  U )  =  (
LSpan `  U )
25 eqid 2253 . . . . . . . . 9  |-  ( +g  `  C )  =  ( +g  `  C )
26 simp11 990 . . . . . . . . . 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 )
2726, 13syl 17 . . . . . . . . 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 ) )
28 hdmap14lem12.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2926, 28syl 17 . . . . . . . . 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.  } ) )
30 simp13 992 . . . . . . . . 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.  } ) )
3126, 18syl 17 . . . . . . . . 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 )
32 hdmap14lem12.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  A )
3326, 32syl 17 . . . . . . . . 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 simp2 961 . . . . . . . . 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 )
35 simp12 991 . . . . . . . . 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 )
) )
361, 2, 3, 22, 4, 23, 24, 5, 6, 7, 25, 8, 10, 11, 12, 27, 29, 30, 31, 33, 34, 35, 21hdmap14lem11 30975 . . . . . . . 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 )
3736oveq1d 5725 . . . . . . 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 )
) )
3821, 37eqtr4d 2288 . . . . . 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 )
) )
3938rexlimdv3a 2631 . . . . 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 ) ) ) )
4020, 39mpd 16 . . . 4  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( S `  ( F  .x.  y
) )  =  ( G  .xb  ( S `  y ) ) )
41403expia 1158 . . 3  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) )  ->  ( y  e.  ( V  \  {  .0.  } )  ->  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
4241ralrimiv 2587 . 2  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) )  ->  A. y  e.  ( V  \  {  .0.  } ) ( S `  ( F  .x.  y ) )  =  ( G 
.xb  ( S `  y ) ) )
43 oveq2 5718 . . . . . . 7  |-  ( y  =  X  ->  ( F  .x.  y )  =  ( F  .x.  X
) )
4443fveq2d 5381 . . . . . 6  |-  ( y  =  X  ->  ( S `  ( F  .x.  y ) )  =  ( S `  ( F  .x.  X ) ) )
45 fveq2 5377 . . . . . . 7  |-  ( y  =  X  ->  ( S `  y )  =  ( S `  X ) )
4645oveq2d 5726 . . . . . 6  |-  ( y  =  X  ->  ( G  .xb  ( S `  y ) )  =  ( G  .xb  ( S `  X )
) )
4744, 46eqeq12d 2267 . . . . 5  |-  ( y  =  X  ->  (
( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  <->  ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) ) ) )
4847rcla4v 2817 . . . 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 ) ) ) )
4928, 48syl 17 . . 3  |-  ( ph  ->  ( A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
)  ->  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) ) )
5049imp 420 . 2  |-  ( (
ph  /\  A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) )  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X )
) )
5142, 50impbida 808 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 set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2509   E.wrex 2510    \ cdif 3075   {csn 3544   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082  Scalarcsca 13085   .scvsca 13086   0gc0g 13274   LSpanclspn 15563   HLchlt 28444   LHypclh 29077   DVecHcdvh 30172  LCDualclcd 30680  HDMapchdma 30887
This theorem is referenced by:  hdmap14lem13  30977
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-ot 3554  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-oppg 14654  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 28070  df-lshyp 28071  df-lcv 28113  df-lfl 28152  df-lkr 28180  df-ldual 28218  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593  df-lines 28594  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198  df-trl 29252  df-tgrp 29836  df-tendo 29848  df-edring 29850  df-dveca 30096  df-disoa 30123  df-dvech 30173  df-dib 30233  df-dic 30267  df-dih 30323  df-doch 30442  df-djh 30489  df-lcdual 30681  df-mapd 30719  df-hvmap 30851  df-hdmap1 30888  df-hdmap 30889
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