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Theorem hdmap14lem13 31240
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
hdmap14lem13  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) )  <->  A. y  e.  V  ( 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 hdmap14lem13
StepHypRef Expression
1 hdmap14lem12.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap14lem12.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap14lem12.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap14lem12.t . . 3  |-  .x.  =  ( .s `  U )
5 hdmap14lem12.r . . 3  |-  R  =  (Scalar `  U )
6 hdmap14lem12.b . . 3  |-  B  =  ( Base `  R
)
7 hdmap14lem12.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap14lem12.e . . 3  |-  .xb  =  ( .s `  C )
9 hdmap14lem12.s . . 3  |-  S  =  ( (HDMap `  K
) `  W )
10 hdmap14lem12.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
11 hdmap14lem12.f . . 3  |-  ( ph  ->  F  e.  B )
12 hdmap14lem12.p . . 3  |-  P  =  (Scalar `  C )
13 hdmap14lem12.a . . 3  |-  A  =  ( Base `  P
)
14 hdmap14lem12.o . . 3  |-  .0.  =  ( 0g `  U )
15 hdmap14lem12.x . . 3  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
16 hdmap14lem12.g . . 3  |-  ( ph  ->  G  e.  A )
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16hdmap14lem12 31239 . 2  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) )  <->  A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
18 elsn 3629 . . . . . 6  |-  ( y  e.  {  .0.  }  <->  y  =  .0.  )
191, 7, 10lcdlmod 30949 . . . . . . . . 9  |-  ( ph  ->  C  e.  LMod )
20 eqid 2258 . . . . . . . . . 10  |-  ( 0g
`  C )  =  ( 0g `  C
)
2112, 8, 13, 20lmodvs0 15626 . . . . . . . . 9  |-  ( ( C  e.  LMod  /\  G  e.  A )  ->  ( G  .xb  ( 0g `  C ) )  =  ( 0g `  C
) )
2219, 16, 21syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( G  .xb  ( 0g `  C ) )  =  ( 0g `  C ) )
231, 2, 14, 7, 20, 9, 10hdmapval0 31193 . . . . . . . . 9  |-  ( ph  ->  ( S `  .0.  )  =  ( 0g `  C ) )
2423oveq2d 5808 . . . . . . . 8  |-  ( ph  ->  ( G  .xb  ( S `  .0.  ) )  =  ( G  .xb  ( 0g `  C ) ) )
251, 2, 10dvhlmod 30467 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
265, 4, 6, 14lmodvs0 15626 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  F  e.  B )  ->  ( F  .x.  .0.  )  =  .0.  )
2725, 11, 26syl2anc 645 . . . . . . . . . 10  |-  ( ph  ->  ( F  .x.  .0.  )  =  .0.  )
2827fveq2d 5462 . . . . . . . . 9  |-  ( ph  ->  ( S `  ( F  .x.  .0.  ) )  =  ( S `  .0.  ) )
2928, 23eqtrd 2290 . . . . . . . 8  |-  ( ph  ->  ( S `  ( F  .x.  .0.  ) )  =  ( 0g `  C ) )
3022, 24, 293eqtr4rd 2301 . . . . . . 7  |-  ( ph  ->  ( S `  ( F  .x.  .0.  ) )  =  ( G  .xb  ( S `  .0.  )
) )
31 oveq2 5800 . . . . . . . . 9  |-  ( y  =  .0.  ->  ( F  .x.  y )  =  ( F  .x.  .0.  ) )
3231fveq2d 5462 . . . . . . . 8  |-  ( y  =  .0.  ->  ( S `  ( F  .x.  y ) )  =  ( S `  ( F  .x.  .0.  ) ) )
33 fveq2 5458 . . . . . . . . 9  |-  ( y  =  .0.  ->  ( S `  y )  =  ( S `  .0.  ) )
3433oveq2d 5808 . . . . . . . 8  |-  ( y  =  .0.  ->  ( G  .xb  ( S `  y ) )  =  ( G  .xb  ( S `  .0.  ) ) )
3532, 34eqeq12d 2272 . . . . . . 7  |-  ( y  =  .0.  ->  (
( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  <->  ( S `  ( F  .x.  .0.  )
)  =  ( G 
.xb  ( S `  .0.  ) ) ) )
3630, 35syl5ibrcom 215 . . . . . 6  |-  ( ph  ->  ( y  =  .0. 
->  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) ) ) )
3718, 36syl5bi 210 . . . . 5  |-  ( ph  ->  ( y  e.  {  .0.  }  ->  ( S `  ( F  .x.  y
) )  =  ( G  .xb  ( S `  y ) ) ) )
3837ralrimiv 2600 . . . 4  |-  ( ph  ->  A. y  e.  {  .0.  }  ( S `  ( F  .x.  y ) )  =  ( G 
.xb  ( S `  y ) ) )
3938biantrud 495 . . 3  |-  ( ph  ->  ( A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
)  <->  ( A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
)  /\  A. y  e.  {  .0.  }  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) ) )
40 ralunb 3331 . . 3  |-  ( A. y  e.  ( ( V  \  {  .0.  }
)  u.  {  .0.  } ) ( S `  ( F  .x.  y ) )  =  ( G 
.xb  ( S `  y ) )  <->  ( A. y  e.  ( V  \  {  .0.  } ) ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  /\  A. y  e.  {  .0.  }  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
4139, 40syl6bbr 256 . 2  |-  ( ph  ->  ( A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
)  <->  A. y  e.  ( ( V  \  {  .0.  } )  u.  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
423, 14lmod0vcl 15621 . . . 4  |-  ( U  e.  LMod  ->  .0.  e.  V )
43 difsnid 3735 . . . 4  |-  (  .0. 
e.  V  ->  (
( V  \  {  .0.  } )  u.  {  .0.  } )  =  V )
4425, 42, 433syl 20 . . 3  |-  ( ph  ->  ( ( V  \  {  .0.  } )  u. 
{  .0.  } )  =  V )
4544raleqdv 2717 . 2  |-  ( ph  ->  ( A. y  e.  ( ( V  \  {  .0.  } )  u. 
{  .0.  } ) ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  <->  A. y  e.  V  ( S `  ( F 
.x.  y ) )  =  ( G  .xb  ( S `  y ) ) ) )
4617, 41, 453bitrd 272 1  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) )  <->  A. y  e.  V  ( S `  ( F  .x.  y
) )  =  ( G  .xb  ( S `  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518    \ cdif 3124    u. cun 3125   {csn 3614   ` cfv 4673  (class class class)co 5792   Basecbs 13110  Scalarcsca 13173   .scvsca 13174   0gc0g 13362   LModclmod 15589   HLchlt 28707   LHypclh 29340   DVecHcdvh 30435  LCDualclcd 30943  HDMapchdma 31150
This theorem is referenced by:  hdmap14lem14  31241
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-ot 3624  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9933  df-z 9992  df-uz 10198  df-fz 10749  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-sca 13186  df-vsca 13187  df-0g 13366  df-mre 13450  df-mrc 13451  df-acs 13453  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-mnd 14329  df-submnd 14378  df-grp 14451  df-minusg 14452  df-sbg 14453  df-subg 14580  df-cntz 14755  df-oppg 14781  df-lsm 14909  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-dvr 15427  df-drng 15476  df-lmod 15591  df-lss 15652  df-lsp 15691  df-lvec 15818  df-lsatoms 28333  df-lshyp 28334  df-lcv 28376  df-lfl 28415  df-lkr 28443  df-ldual 28481  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tgrp 30099  df-tendo 30111  df-edring 30113  df-dveca 30359  df-disoa 30386  df-dvech 30436  df-dib 30496  df-dic 30530  df-dih 30586  df-doch 30705  df-djh 30752  df-lcdual 30944  df-mapd 30982  df-hvmap 31114  df-hdmap1 31151  df-hdmap 31152
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