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Theorem hdmap14lem8 32361
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 32075 . . 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 2404 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
9 hdmap14lem8.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
10 hdmap14lem8.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
1110eldifad 3292 . . . 4  |-  ( ph  ->  X  e.  V )
121, 6, 7, 2, 8, 9, 3, 11hdmapcl 32316 . . 3  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
13 hdmap14lem8.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3292 . . . 4  |-  ( ph  ->  Y  e.  V )
151, 6, 7, 2, 8, 9, 3, 14hdmapcl 32316 . . 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 15928 . . 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 32329 . . . 4  |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( ( S `
 X )  .+b  ( S `  Y ) ) )
2423oveq2d 6056 . . 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 31593 . . . . . . 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 15928 . . . . . . 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 5691 . . . . 5  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( S `  ( ( F  .x.  X )  .+  ( F  .x.  Y ) ) ) )
347, 28, 29, 30lmodvscl 15922 . . . . . . 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 15922 . . . . . . 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 32329 . . . . 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 6058 . . . . 5  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  .+b  ( S `  ( F  .x.  Y
) ) )  =  ( ( G  .xb  ( S `  X ) )  .+b  ( I  .xb  ( S `  Y
) ) ) )
4233, 38, 413eqtrd 2440 . . . 4  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4325, 42eqtr3d 2438 . . 3  |-  ( ph  ->  ( J  .xb  ( S `  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4424, 43eqtr3d 2438 . 2  |-  ( ph  ->  ( J  .xb  (
( S `  X
)  .+b  ( S `  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4521, 44eqtr3d 2438 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 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277   {csn 3774   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   0gc0g 13678   LModclmod 15905   LSpanclspn 16002   HLchlt 29833   LHypclh 30466   DVecHcdvh 31561  LCDualclcd 32069  HDMapchdma 32276
This theorem is referenced by:  hdmap14lem9  32362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lshyp 29460  df-lcv 29502  df-lfl 29541  df-lkr 29569  df-ldual 29607  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tgrp 31225  df-tendo 31237  df-edring 31239  df-dveca 31485  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831  df-djh 31878  df-lcdual 32070  df-mapd 32108  df-hvmap 32240  df-hdmap1 32277  df-hdmap 32278
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