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Theorem hdmap14lem8 31335
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 31049 . . 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 2284 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
9 hdmap14lem8.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
10 hdmap14lem8.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
11 eldifi 3299 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
1210, 11syl 17 . . . 4  |-  ( ph  ->  X  e.  V )
131, 6, 7, 2, 8, 9, 3, 12hdmapcl 31290 . . 3  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
14 hdmap14lem8.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
15 eldifi 3299 . . . . 5  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1614, 15syl 17 . . . 4  |-  ( ph  ->  Y  e.  V )
171, 6, 7, 2, 8, 9, 3, 16hdmapcl 31290 . . 3  |-  ( ph  ->  ( S `  Y
)  e.  ( Base `  C ) )
18 hdmap14lem8.d . . . 4  |-  .+b  =  ( +g  `  C )
19 hdmap14lem8.p . . . 4  |-  P  =  (Scalar `  C )
20 hdmap14lem8.e . . . 4  |-  .xb  =  ( .s `  C )
21 hdmap14lem8.a . . . 4  |-  A  =  ( Base `  P
)
228, 18, 19, 20, 21lmodvsdi 15644 . . 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
) ) ) )
234, 5, 13, 17, 22syl13anc 1189 . 2  |-  ( ph  ->  ( J  .xb  (
( S `  X
)  .+b  ( S `  Y ) ) )  =  ( ( J 
.xb  ( S `  X ) )  .+b  ( J  .xb  ( S `
 Y ) ) ) )
24 hdmap14lem8.q . . . . 5  |-  .+  =  ( +g  `  U )
251, 6, 7, 24, 2, 18, 9, 3, 12, 16hdmapadd 31303 . . . 4  |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( ( S `
 X )  .+b  ( S `  Y ) ) )
2625oveq2d 5835 . . 3  |-  ( ph  ->  ( J  .xb  ( S `  ( X  .+  Y ) ) )  =  ( J  .xb  ( ( S `  X )  .+b  ( S `  Y )
) ) )
27 hdmap14lem8.xy . . . 4  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( J  .xb  ( S `  ( X 
.+  Y ) ) ) )
281, 6, 3dvhlmod 30567 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
29 hdmap14lem8.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
30 hdmap14lem8.r . . . . . . . 8  |-  R  =  (Scalar `  U )
31 hdmap14lem8.t . . . . . . . 8  |-  .x.  =  ( .s `  U )
32 hdmap14lem8.b . . . . . . . 8  |-  B  =  ( Base `  R
)
337, 24, 30, 31, 32lmodvsdi 15644 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( F  e.  B  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( F  .x.  ( X  .+  Y
) )  =  ( ( F  .x.  X
)  .+  ( F  .x.  Y ) ) )
3428, 29, 12, 16, 33syl13anc 1189 . . . . . 6  |-  ( ph  ->  ( F  .x.  ( X  .+  Y ) )  =  ( ( F 
.x.  X )  .+  ( F  .x.  Y ) ) )
3534fveq2d 5489 . . . . 5  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( S `  ( ( F  .x.  X )  .+  ( F  .x.  Y ) ) ) )
367, 30, 31, 32lmodvscl 15638 . . . . . . 7  |-  ( ( U  e.  LMod  /\  F  e.  B  /\  X  e.  V )  ->  ( F  .x.  X )  e.  V )
3728, 29, 12, 36syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( F  .x.  X
)  e.  V )
387, 30, 31, 32lmodvscl 15638 . . . . . . 7  |-  ( ( U  e.  LMod  /\  F  e.  B  /\  Y  e.  V )  ->  ( F  .x.  Y )  e.  V )
3928, 29, 16, 38syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( F  .x.  Y
)  e.  V )
401, 6, 7, 24, 2, 18, 9, 3, 37, 39hdmapadd 31303 . . . . 5  |-  ( ph  ->  ( S `  (
( F  .x.  X
)  .+  ( F  .x.  Y ) ) )  =  ( ( S `
 ( F  .x.  X ) )  .+b  ( S `  ( F 
.x.  Y ) ) ) )
41 hdmap14lem8.xx . . . . . 6  |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )
42 hdmap14lem8.yy . . . . . 6  |-  ( ph  ->  ( S `  ( F  .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )
4341, 42oveq12d 5837 . . . . 5  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  .+b  ( S `  ( F  .x.  Y
) ) )  =  ( ( G  .xb  ( S `  X ) )  .+b  ( I  .xb  ( S `  Y
) ) ) )
4435, 40, 433eqtrd 2320 . . . 4  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4527, 44eqtr3d 2318 . . 3  |-  ( ph  ->  ( J  .xb  ( S `  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4626, 45eqtr3d 2318 . 2  |-  ( ph  ->  ( J  .xb  (
( S `  X
)  .+b  ( S `  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4723, 46eqtr3d 2318 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 6    /\ wa 360    = wceq 1628    e. wcel 1688    =/= wne 2447    \ cdif 3150   {csn 3641   ` cfv 5221  (class class class)co 5819   Basecbs 13142   +g cplusg 13202  Scalarcsca 13205   .scvsca 13206   0gc0g 13394   LModclmod 15621   LSpanclspn 15722   HLchlt 28807   LHypclh 29440   DVecHcdvh 30535  LCDualclcd 31043  HDMapchdma 31250
This theorem is referenced by:  hdmap14lem9  31336
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-ot 3651  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-undef 6291  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-n0 9961  df-z 10020  df-uz 10226  df-fz 10777  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-sca 13218  df-vsca 13219  df-0g 13398  df-mre 13482  df-mrc 13483  df-acs 13485  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-mnd 14361  df-submnd 14410  df-grp 14483  df-minusg 14484  df-sbg 14485  df-subg 14612  df-cntz 14787  df-oppg 14813  df-lsm 14941  df-cmn 15085  df-abl 15086  df-mgp 15320  df-rng 15334  df-ur 15336  df-oppr 15399  df-dvdsr 15417  df-unit 15418  df-invr 15448  df-dvr 15459  df-drng 15508  df-lmod 15623  df-lss 15684  df-lsp 15723  df-lvec 15850  df-lsatoms 28433  df-lshyp 28434  df-lcv 28476  df-lfl 28515  df-lkr 28543  df-ldual 28581  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tgrp 30199  df-tendo 30211  df-edring 30213  df-dveca 30459  df-disoa 30486  df-dvech 30536  df-dib 30596  df-dic 30630  df-dih 30686  df-doch 30805  df-djh 30852  df-lcdual 31044  df-mapd 31082  df-hvmap 31214  df-hdmap1 31251  df-hdmap 31252
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