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Theorem hdmap14lem12 37710
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 2457 . . . . . 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 1017 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 simp3 998 . . . . . . 7  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  y  e.  ( V  \  {  .0.  } ) )
1615eldifad 3483 . . . . . 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 1017 . . . . . 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 37698 . . . . 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 998 . . . . . . 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 2457 . . . . . . . . 9  |-  ( +g  `  U )  =  ( +g  `  U )
22 hdmap14lem12.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
23 eqid 2457 . . . . . . . . 9  |-  ( LSpan `  U )  =  (
LSpan `  U )
24 eqid 2457 . . . . . . . . 9  |-  ( +g  `  C )  =  ( +g  `  C )
25 simp11 1026 . . . . . . . . . 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 1028 . . . . . . . . 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 997 . . . . . . . . 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 1027 . . . . . . . . 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 37709 . . . . . . . 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 6311 . . . . . . 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 2501 . . . . . 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 2951 . . . . 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 1198 . . 3  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) )  ->  ( y  e.  ( V  \  {  .0.  } )  ->  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
4140ralrimiv 2869 . 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 6304 . . . . . . 7  |-  ( y  =  X  ->  ( F  .x.  y )  =  ( F  .x.  X
) )
4342fveq2d 5876 . . . . . 6  |-  ( y  =  X  ->  ( S `  ( F  .x.  y ) )  =  ( S `  ( F  .x.  X ) ) )
44 fveq2 5872 . . . . . . 7  |-  ( y  =  X  ->  ( S `  y )  =  ( S `  X ) )
4544oveq2d 6312 . . . . . 6  |-  ( y  =  X  ->  ( G  .xb  ( S `  y ) )  =  ( G  .xb  ( S `  X )
) )
4643, 45eqeq12d 2479 . . . . 5  |-  ( y  =  X  ->  (
( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  <->  ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) ) ) )
4746rspcv 3206 . . . 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 832 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    \ cdif 3468   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711  Scalarcsca 14714   .scvsca 14715   0gc0g 14856   LSpanclspn 17743   HLchlt 35176   LHypclh 35809   DVecHcdvh 36906  LCDualclcd 37414  HDMapchdma 37621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-undef 7020  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-0g 14858  df-mre 15002  df-mrc 15003  df-acs 15005  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-oppg 16507  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-dvr 17458  df-drng 17524  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lvec 17875  df-lsatoms 34802  df-lshyp 34803  df-lcv 34845  df-lfl 34884  df-lkr 34912  df-ldual 34950  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985  df-tgrp 36570  df-tendo 36582  df-edring 36584  df-dveca 36830  df-disoa 36857  df-dvech 36907  df-dib 36967  df-dic 37001  df-dih 37057  df-doch 37176  df-djh 37223  df-lcdual 37415  df-mapd 37453  df-hvmap 37585  df-hdmap1 37622  df-hdmap 37623
This theorem is referenced by:  hdmap14lem13  37711
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