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Theorem hdmap14lem12 31202
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 2256 . . . . . 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 3240 . . . . . . 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 31190 . . . . 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 2256 . . . . . . . . 9  |-  ( +g  `  U )  =  ( +g  `  U )
23 hdmap14lem12.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
24 eqid 2256 . . . . . . . . 9  |-  ( LSpan `  U )  =  (
LSpan `  U )
25 eqid 2256 . . . . . . . . 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 31201 . . . . . . . 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 5772 . . . . . . 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 2291 . . . . . 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 2640 . . . . 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 2596 . 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 5765 . . . . . . 7  |-  ( y  =  X  ->  ( F  .x.  y )  =  ( F  .x.  X
) )
4443fveq2d 5427 . . . . . 6  |-  ( y  =  X  ->  ( S `  ( F  .x.  y ) )  =  ( S `  ( F  .x.  X ) ) )
45 fveq2 5423 . . . . . . 7  |-  ( y  =  X  ->  ( S `  y )  =  ( S `  X ) )
4645oveq2d 5773 . . . . . 6  |-  ( y  =  X  ->  ( G  .xb  ( S `  y ) )  =  ( G  .xb  ( S `  X )
) )
4744, 46eqeq12d 2270 . . . . 5  |-  ( y  =  X  ->  (
( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  <->  ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) ) ) )
4847rcla4v 2831 . . . 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 2516   E.wrex 2517    \ cdif 3091   {csn 3581   ` cfv 4638  (class class class)co 5757   Basecbs 13075   +g cplusg 13135  Scalarcsca 13138   .scvsca 13139   0gc0g 13327   LSpanclspn 15655   HLchlt 28670   LHypclh 29303   DVecHcdvh 30398  LCDualclcd 30906  HDMapchdma 31113
This theorem is referenced by:  hdmap14lem13  31203
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-ot 3591  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-tpos 6133  df-iota 6190  df-undef 6229  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-n0 9898  df-z 9957  df-uz 10163  df-fz 10714  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-sca 13151  df-vsca 13152  df-0g 13331  df-mre 13415  df-mrc 13416  df-acs 13418  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-mnd 14294  df-submnd 14343  df-grp 14416  df-minusg 14417  df-sbg 14418  df-subg 14545  df-cntz 14720  df-oppg 14746  df-lsm 14874  df-cmn 15018  df-abl 15019  df-mgp 15253  df-ring 15267  df-ur 15269  df-oppr 15332  df-dvdsr 15350  df-unit 15351  df-invr 15381  df-dvr 15392  df-drng 15441  df-lmod 15556  df-lss 15617  df-lsp 15656  df-lvec 15783  df-lsatoms 28296  df-lshyp 28297  df-lcv 28339  df-lfl 28378  df-lkr 28406  df-ldual 28444  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478  df-tgrp 30062  df-tendo 30074  df-edring 30076  df-dveca 30322  df-disoa 30349  df-dvech 30399  df-dib 30459  df-dic 30493  df-dih 30549  df-doch 30668  df-djh 30715  df-lcdual 30907  df-mapd 30945  df-hvmap 31077  df-hdmap1 31114  df-hdmap 31115
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