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Theorem hdmap14lem8 31235
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 30949 . . 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 2258 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
9 hdmap14lem8.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
10 hdmap14lem8.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
11 eldifi 3273 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
1210, 11syl 17 . . . 4  |-  ( ph  ->  X  e.  V )
131, 6, 7, 2, 8, 9, 3, 12hdmapcl 31190 . . 3  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  C ) )
14 hdmap14lem8.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
15 eldifi 3273 . . . . 5  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1614, 15syl 17 . . . 4  |-  ( ph  ->  Y  e.  V )
171, 6, 7, 2, 8, 9, 3, 16hdmapcl 31190 . . 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 15612 . . 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 31203 . . . 4  |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( ( S `
 X )  .+b  ( S `  Y ) ) )
2625oveq2d 5808 . . 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 30467 . . . . . . 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 15612 . . . . . . 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 5462 . . . . 5  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( S `  ( ( F  .x.  X )  .+  ( F  .x.  Y ) ) ) )
367, 30, 31, 32lmodvscl 15606 . . . . . . 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 15606 . . . . . . 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 31203 . . . . 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 5810 . . . . 5  |-  ( ph  ->  ( ( S `  ( F  .x.  X ) )  .+b  ( S `  ( F  .x.  Y
) ) )  =  ( ( G  .xb  ( S `  X ) )  .+b  ( I  .xb  ( S `  Y
) ) ) )
4435, 40, 433eqtrd 2294 . . . 4  |-  ( ph  ->  ( S `  ( F  .x.  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4527, 44eqtr3d 2292 . . 3  |-  ( ph  ->  ( J  .xb  ( S `  ( X  .+  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4626, 45eqtr3d 2292 . 2  |-  ( ph  ->  ( J  .xb  (
( S `  X
)  .+b  ( S `  Y ) ) )  =  ( ( G 
.xb  ( S `  X ) )  .+b  ( I  .xb  ( S `
 Y ) ) ) )
4723, 46eqtr3d 2292 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 1619    e. wcel 1621    =/= wne 2421    \ cdif 3124   {csn 3614   ` cfv 4673  (class class class)co 5792   Basecbs 13110   +g cplusg 13170  Scalarcsca 13173   .scvsca 13174   0gc0g 13362   LModclmod 15589   LSpanclspn 15690   HLchlt 28707   LHypclh 29340   DVecHcdvh 30435  LCDualclcd 30943  HDMapchdma 31150
This theorem is referenced by:  hdmap14lem9  31236
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-ot 3624  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9933  df-z 9992  df-uz 10198  df-fz 10749  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-sca 13186  df-vsca 13187  df-0g 13366  df-mre 13450  df-mrc 13451  df-acs 13453  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-mnd 14329  df-submnd 14378  df-grp 14451  df-minusg 14452  df-sbg 14453  df-subg 14580  df-cntz 14755  df-oppg 14781  df-lsm 14909  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-dvr 15427  df-drng 15476  df-lmod 15591  df-lss 15652  df-lsp 15691  df-lvec 15818  df-lsatoms 28333  df-lshyp 28334  df-lcv 28376  df-lfl 28415  df-lkr 28443  df-ldual 28481  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tgrp 30099  df-tendo 30111  df-edring 30113  df-dveca 30359  df-disoa 30386  df-dvech 30436  df-dib 30496  df-dic 30530  df-dih 30586  df-doch 30705  df-djh 30752  df-lcdual 30944  df-mapd 30982  df-hvmap 31114  df-hdmap1 31151  df-hdmap 31152
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