MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lvecindp Unicode version

Theorem lvecindp 15891
Description: Compute the  X coefficient in a sum with an independent vector  X (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions  Y and 
Z (second conjunct). Typically  U is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
lvecindp.v  |-  V  =  ( Base `  W
)
lvecindp.p  |-  .+  =  ( +g  `  W )
lvecindp.f  |-  F  =  (Scalar `  W )
lvecindp.k  |-  K  =  ( Base `  F
)
lvecindp.t  |-  .x.  =  ( .s `  W )
lvecindp.s  |-  S  =  ( LSubSp `  W )
lvecindp.w  |-  ( ph  ->  W  e.  LVec )
lvecindp.u  |-  ( ph  ->  U  e.  S )
lvecindp.x  |-  ( ph  ->  X  e.  V )
lvecindp.n  |-  ( ph  ->  -.  X  e.  U
)
lvecindp.y  |-  ( ph  ->  Y  e.  U )
lvecindp.z  |-  ( ph  ->  Z  e.  U )
lvecindp.a  |-  ( ph  ->  A  e.  K )
lvecindp.b  |-  ( ph  ->  B  e.  K )
lvecindp.e  |-  ( ph  ->  ( ( A  .x.  X )  .+  Y
)  =  ( ( B  .x.  X ) 
.+  Z ) )
Assertion
Ref Expression
lvecindp  |-  ( ph  ->  ( A  =  B  /\  Y  =  Z ) )

Proof of Theorem lvecindp
StepHypRef Expression
1 lvecindp.p . . . 4  |-  .+  =  ( +g  `  W )
2 eqid 2283 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
3 eqid 2283 . . . 4  |-  (Cntz `  W )  =  (Cntz `  W )
4 lvecindp.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 15859 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 15 . . . . 5  |-  ( ph  ->  W  e.  LMod )
7 lvecindp.x . . . . 5  |-  ( ph  ->  X  e.  V )
8 lvecindp.v . . . . . 6  |-  V  =  ( Base `  W
)
9 eqid 2283 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
108, 9lspsnsubg 15737 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( LSpan `  W ) `  { X } )  e.  (SubGrp `  W
) )
116, 7, 10syl2anc 642 . . . 4  |-  ( ph  ->  ( ( LSpan `  W
) `  { X } )  e.  (SubGrp `  W ) )
12 lvecindp.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
1312lsssssubg 15715 . . . . . 6  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
146, 13syl 15 . . . . 5  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
15 lvecindp.u . . . . 5  |-  ( ph  ->  U  e.  S )
1614, 15sseldd 3181 . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
17 lvecindp.n . . . . 5  |-  ( ph  ->  -.  X  e.  U
)
188, 2, 9, 12, 4, 15, 7, 17lspdisj 15878 . . . 4  |-  ( ph  ->  ( ( ( LSpan `  W ) `  { X } )  i^i  U
)  =  { ( 0g `  W ) } )
19 lmodabl 15672 . . . . . 6  |-  ( W  e.  LMod  ->  W  e. 
Abel )
206, 19syl 15 . . . . 5  |-  ( ph  ->  W  e.  Abel )
213, 20, 11, 16ablcntzd 15149 . . . 4  |-  ( ph  ->  ( ( LSpan `  W
) `  { X } )  C_  (
(Cntz `  W ) `  U ) )
22 lvecindp.t . . . . 5  |-  .x.  =  ( .s `  W )
23 lvecindp.f . . . . 5  |-  F  =  (Scalar `  W )
24 lvecindp.k . . . . 5  |-  K  =  ( Base `  F
)
25 lvecindp.a . . . . 5  |-  ( ph  ->  A  e.  K )
268, 22, 23, 24, 9, 6, 25, 7lspsneli 15758 . . . 4  |-  ( ph  ->  ( A  .x.  X
)  e.  ( (
LSpan `  W ) `  { X } ) )
27 lvecindp.b . . . . 5  |-  ( ph  ->  B  e.  K )
288, 22, 23, 24, 9, 6, 27, 7lspsneli 15758 . . . 4  |-  ( ph  ->  ( B  .x.  X
)  e.  ( (
LSpan `  W ) `  { X } ) )
29 lvecindp.y . . . 4  |-  ( ph  ->  Y  e.  U )
30 lvecindp.z . . . 4  |-  ( ph  ->  Z  e.  U )
31 lvecindp.e . . . 4  |-  ( ph  ->  ( ( A  .x.  X )  .+  Y
)  =  ( ( B  .x.  X ) 
.+  Z ) )
321, 2, 3, 11, 16, 18, 21, 26, 28, 29, 30, 31subgdisj1 15000 . . 3  |-  ( ph  ->  ( A  .x.  X
)  =  ( B 
.x.  X ) )
338, 2, 12, 6, 15, 7, 17lssneln0 15709 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  { ( 0g
`  W ) } ) )
34 eldifsni 3750 . . . . 5  |-  ( X  e.  ( V  \  { ( 0g `  W ) } )  ->  X  =/=  ( 0g `  W ) )
3533, 34syl 15 . . . 4  |-  ( ph  ->  X  =/=  ( 0g
`  W ) )
368, 22, 23, 24, 2, 4, 25, 27, 7, 35lvecvscan2 15865 . . 3  |-  ( ph  ->  ( ( A  .x.  X )  =  ( B  .x.  X )  <-> 
A  =  B ) )
3732, 36mpbid 201 . 2  |-  ( ph  ->  A  =  B )
381, 2, 3, 11, 16, 18, 21, 26, 28, 29, 30, 31subgdisj2 15001 . 2  |-  ( ph  ->  Y  =  Z )
3937, 38jca 518 1  |-  ( ph  ->  ( A  =  B  /\  Y  =  Z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400  SubGrpcsubg 14615  Cntzccntz 14791   Abelcabel 15090   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855
This theorem is referenced by:  baerlem3lem1  31270  baerlem5alem1  31271  baerlem5blem1  31272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856
  Copyright terms: Public domain W3C validator