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Theorem hdmap14lem8 32515
Description: Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.)
Hypotheses
Ref Expression
hdmap14lem8.h  |-  H  =  ( LHyp `  K
)
hdmap14lem8.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap14lem8.v  |-  V  =  ( Base `  U
)
hdmap14lem8.q  |-  .+  =  ( +g  `  U )
hdmap14lem8.t  |-  .x.  =  ( .s `  U )
hdmap14lem8.o  |-  .0.  =  ( 0g `  U )
hdmap14lem8.n  |-  N  =  ( LSpan `  U )
hdmap14lem8.r  |-  R  =  (Scalar `  U )
hdmap14lem8.b  |-  B  =  ( Base `  R
)
hdmap14lem8.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap14lem8.d  |-  .+b  =  ( +g  `  C )
hdmap14lem8.e  |-  .xb  =  ( .s `  C )
hdmap14lem8.p  |-  P  =  (Scalar `  C )
hdmap14lem8.a  |-  A  =  ( Base `  P
)
hdmap14lem8.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmap14lem8.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap14lem8.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap14lem8.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
hdmap14lem8.f  |-  ( ph  ->  F  e.  B )
hdmap14lem8.g  |-  ( ph  ->  G  e.  A )
hdmap14lem8.i  |-  ( ph  ->  I  e.  A )
hdmap14lem8.xx  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )
hdmap14lem8.yy  |-  ( ph  ->  ( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )
hdmap14lem8.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
hdmap14lem8.j  |-  ( ph  ->  J  e.  A )
hdmap14lem8.xy  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( J  .xb  ( S `  ( X 
.+  Y ) ) ) )
Assertion
Ref Expression
hdmap14lem8  |-  ( ph  ->  ( ( J  .xb  ( S `  X ) )  .+b  ( J  .xb  ( S `  Y
) ) )  =  ( ( G  .xb  ( S `  X ) )  .+b  ( I  .xb  ( S `  Y
) ) ) )

Proof of Theorem hdmap14lem8
StepHypRef Expression
1 hdmap14lem8.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmap14lem8.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
3 hdmap14lem8.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3lcdlmod 32229 . . 3  |-  ( ph  ->  C  e.  LMod )
5 hdmap14lem8.j . . 3  |-  ( ph  ->  J  e.  A )
6 hdmap14lem8.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
7 hdmap14lem8.v . . . 4  |-  V  =  ( Base `  U
)
8 eqid 2435 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
9 hdmap14lem8.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
10 hdmap14lem8.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
1110eldifad 3324 . . . 4  |-  ( ph  ->  X  e.  V )
121, 6, 7, 2, 8, 9, 3, 11hdmapcl 32470 . . 3  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
13 hdmap14lem8.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3324 . . . 4  |-  ( ph  ->  Y  e.  V )
151, 6, 7, 2, 8, 9, 3, 14hdmapcl 32470 . . 3  |-  ( ph  ->  ( S `  Y
)  e.  ( Base `  C ) )
16 hdmap14lem8.d . . . 4  |-  .+b  =  ( +g  `  C )
17 hdmap14lem8.p . . . 4  |-  P  =  (Scalar `  C )
18 hdmap14lem8.e . . . 4  |-  .xb  =  ( .s `  C )
19 hdmap14lem8.a . . . 4  |-  A  =  ( Base `  P
)
208, 16, 17, 18, 19lmodvsdi 15961 . . 3  |-  ( ( C  e.  LMod  /\  ( J  e.  A  /\  ( S `  X )  e.  ( Base `  C
)  /\  ( S `  Y )  e.  (
Base `  C )
) )  ->  ( J  .xb  ( ( S `
 X )  .+b  ( S `  Y ) ) )  =  ( ( J  .xb  ( S `  X )
)  .+b  ( J  .xb  ( S `  Y
) ) ) )
214, 5, 12, 15, 20syl13anc 1186 . 2  |-  ( ph  ->  ( J  .xb  (
( S `  X
)  .+b  ( S `  Y ) ) )  =  ( ( J 
.xb  ( S `  X ) )  .+b  ( J  .xb  ( S `
 Y ) ) ) )
22 hdmap14lem8.q . . . . 5  |-  .+  =  ( +g  `  U )
231, 6, 7, 22, 2, 16, 9, 3, 11, 14hdmapadd 32483 . . . 4  |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( ( S `
 X )  .+b  ( S `  Y ) ) )
2423oveq2d 6088 . . 3  |-  ( ph  ->  ( J  .xb  ( S `  ( X  .+  Y ) ) )  =  ( J  .xb  ( ( S `  X )  .+b  ( S `  Y )
) ) )
25 hdmap14lem8.xy . . . 4  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( J  .xb  ( S `  ( X 
.+  Y ) ) ) )
261, 6, 3dvhlmod 31747 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
27 hdmap14lem8.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
28 hdmap14lem8.r . . . . . . . 8  |-  R  =  (Scalar `  U )
29 hdmap14lem8.t . . . . . . . 8  |-  .x.  =  ( .s `  U )
30 hdmap14lem8.b . . . . . . . 8  |-  B  =  ( Base `  R
)
317, 22, 28, 29, 30lmodvsdi 15961 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( F  e.  B  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( F  .x.  ( X  .+  Y
) )  =  ( ( F  .x.  X
)  .+  ( F  .x.  Y ) ) )
3226, 27, 11, 14, 31syl13anc 1186 . . . . . 6  |-  ( ph  ->  ( F  .x.  ( X  .+  Y ) )  =  ( ( F 
.x.  X )  .+  ( F  .x.  Y ) ) )
3332fveq2d 5723 . . . . 5  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( S `  ( ( F  .x.  X )  .+  ( F  .x.  Y ) ) ) )
347, 28, 29, 30lmodvscl 15955 . . . . . . 7  |-  ( ( U  e.  LMod  /\  F  e.  B  /\  X  e.  V )  ->  ( F  .x.  X )  e.  V )
3526, 27, 11, 34syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( F  .x.  X
)  e.  V )
367, 28, 29, 30lmodvscl 15955 . . . . . . 7  |-  ( ( U  e.  LMod  /\  F  e.  B  /\  Y  e.  V )  ->  ( F  .x.  Y )  e.  V )
3726, 27, 14, 36syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( F  .x.  Y
)  e.  V )
381, 6, 7, 22, 2, 16, 9, 3, 35, 37hdmapadd 32483 . . . . 5  |-  ( ph  ->  ( S `  (
( F  .x.  X
)  .+  ( F  .x.  Y ) ) )  =  ( ( S `
 ( F  .x.  X ) )  .+b  ( S `  ( F 
.x.  Y ) ) ) )
39 hdmap14lem8.xx . . . . . 6  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )
40 hdmap14lem8.yy . . . . . 6  |-  ( ph  ->  ( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )
4139, 40oveq12d 6090 . . . . 5  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  .+b  ( S `  ( F  .x.  Y
) ) )  =  ( ( G  .xb  ( S `  X ) )  .+b  ( I  .xb  ( S `  Y
) ) ) )
4233, 38, 413eqtrd 2471 . . . 4  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4325, 42eqtr3d 2469 . . 3  |-  ( ph  ->  ( J  .xb  ( S `  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4424, 43eqtr3d 2469 . 2  |-  ( ph  ->  ( J  .xb  (
( S `  X
)  .+b  ( S `  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4521, 44eqtr3d 2469 1  |-  ( ph  ->  ( ( J  .xb  ( S `  X ) )  .+b  ( J  .xb  ( S `  Y
) ) )  =  ( ( G  .xb  ( S `  X ) )  .+b  ( I  .xb  ( S `  Y
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   ` cfv 5445  (class class class)co 6072   Basecbs 13457   +g cplusg 13517  Scalarcsca 13520   .scvsca 13521   0gc0g 13711   LModclmod 15938   LSpanclspn 16035   HLchlt 29987   LHypclh 30620   DVecHcdvh 31715  LCDualclcd 32223  HDMapchdma 32430
This theorem is referenced by:  hdmap14lem9  32516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-tpos 6470  df-undef 6534  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-0g 13715  df-mre 13799  df-mrc 13800  df-acs 13802  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-subg 14929  df-cntz 15104  df-oppg 15130  df-lsm 15258  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-ur 15653  df-oppr 15716  df-dvdsr 15734  df-unit 15735  df-invr 15765  df-dvr 15776  df-drng 15825  df-lmod 15940  df-lss 15997  df-lsp 16036  df-lvec 16163  df-lsatoms 29613  df-lshyp 29614  df-lcv 29656  df-lfl 29695  df-lkr 29723  df-ldual 29761  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tgrp 31379  df-tendo 31391  df-edring 31393  df-dveca 31639  df-disoa 31666  df-dvech 31716  df-dib 31776  df-dic 31810  df-dih 31866  df-doch 31985  df-djh 32032  df-lcdual 32224  df-mapd 32262  df-hvmap 32394  df-hdmap1 32431  df-hdmap 32432
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