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Theorem hdmap14lem12 32145
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 2285 . . . . . 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 976 . . . . . 6  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 simp3 957 . . . . . . 7  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  y  e.  ( V  \  {  .0.  } ) )
16 eldifi 3300 . . . . . . 7  |-  ( y  e.  ( V  \  {  .0.  } )  -> 
y  e.  V )
1715, 16syl 15 . . . . . 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 976 . . . . . 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 32133 . . . . 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 957 . . . . . . 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 2285 . . . . . . . . 9  |-  ( +g  `  U )  =  ( +g  `  U )
23 hdmap14lem12.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
24 eqid 2285 . . . . . . . . 9  |-  ( LSpan `  U )  =  (
LSpan `  U )
25 eqid 2285 . . . . . . . . 9  |-  ( +g  `  C )  =  ( +g  `  C )
26 simp11 985 . . . . . . . . . 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 15 . . . . . . . . 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 15 . . . . . . . . 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 987 . . . . . . . . 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 15 . . . . . . . . 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 15 . . . . . . . . 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 956 . . . . . . . . 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 986 . . . . . . . . 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 32144 . . . . . . . 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 5875 . . . . . . 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 2320 . . . . . 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 2671 . . . . 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 14 . . . 4  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) )  /\  y  e.  ( V  \  {  .0.  } ) )  ->  ( S `  ( F  .x.  y
) )  =  ( G  .xb  ( S `  y ) ) )
41403expia 1153 . . 3  |-  ( (
ph  /\  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) )  ->  ( y  e.  ( V  \  {  .0.  } )  ->  ( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
) ) )
4241ralrimiv 2627 . 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 5868 . . . . . . 7  |-  ( y  =  X  ->  ( F  .x.  y )  =  ( F  .x.  X
) )
4443fveq2d 5531 . . . . . 6  |-  ( y  =  X  ->  ( S `  ( F  .x.  y ) )  =  ( S `  ( F  .x.  X ) ) )
45 fveq2 5527 . . . . . . 7  |-  ( y  =  X  ->  ( S `  y )  =  ( S `  X ) )
4645oveq2d 5876 . . . . . 6  |-  ( y  =  X  ->  ( G  .xb  ( S `  y ) )  =  ( G  .xb  ( S `  X )
) )
4744, 46eqeq12d 2299 . . . . 5  |-  ( y  =  X  ->  (
( S `  ( F  .x.  y ) )  =  ( G  .xb  ( S `  y ) )  <->  ( S `  ( F  .x.  X ) )  =  ( G 
.xb  ( S `  X ) ) ) )
4847rspcv 2882 . . . 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 15 . . 3  |-  ( ph  ->  ( A. y  e.  ( V  \  {  .0.  } ) ( S `
 ( F  .x.  y ) )  =  ( G  .xb  ( S `  y )
)  ->  ( S `  ( F  .x.  X
) )  =  ( G  .xb  ( S `  X ) ) ) )
5049imp 418 . 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 805 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 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546    \ cdif 3151   {csn 3642   ` cfv 5257  (class class class)co 5860   Basecbs 13150   +g cplusg 13210  Scalarcsca 13213   .scvsca 13214   0gc0g 13402   LSpanclspn 15730   HLchlt 29613   LHypclh 30246   DVecHcdvh 31341  LCDualclcd 31849  HDMapchdma 32056
This theorem is referenced by:  hdmap14lem13  32146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-ot 3652  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-mre 13490  df-mrc 13491  df-acs 13493  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-cntz 14795  df-oppg 14821  df-lsm 14949  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-lsatoms 29239  df-lshyp 29240  df-lcv 29282  df-lfl 29321  df-lkr 29349  df-ldual 29387  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tgrp 31005  df-tendo 31017  df-edring 31019  df-dveca 31265  df-disoa 31292  df-dvech 31342  df-dib 31402  df-dic 31436  df-dih 31492  df-doch 31611  df-djh 31658  df-lcdual 31850  df-mapd 31888  df-hvmap 32020  df-hdmap1 32057  df-hdmap 32058
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