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Theorem lvecindp 15907
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 2296 . . . 4  |-  ( 0g
`  W )  =  ( 0g `  W
)
3 eqid 2296 . . . 4  |-  (Cntz `  W )  =  (Cntz `  W )
4 lvecindp.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 15875 . . . . . 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 2296 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
108, 9lspsnsubg 15753 . . . . 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 15731 . . . . . 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 3194 . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
17 lvecindp.n . . . . 5  |-  ( ph  ->  -.  X  e.  U
)
188, 2, 9, 12, 4, 15, 7, 17lspdisj 15894 . . . 4  |-  ( ph  ->  ( ( ( LSpan `  W ) `  { X } )  i^i  U
)  =  { ( 0g `  W ) } )
19 lmodabl 15688 . . . . . 6  |-  ( W  e.  LMod  ->  W  e. 
Abel )
206, 19syl 15 . . . . 5  |-  ( ph  ->  W  e.  Abel )
213, 20, 11, 16ablcntzd 15165 . . . 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 15774 . . . 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 15774 . . . 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 15016 . . 3  |-  ( ph  ->  ( A  .x.  X
)  =  ( B 
.x.  X ) )
338, 2, 12, 6, 15, 7, 17lssneln0 15725 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  { ( 0g
`  W ) } ) )
34 eldifsni 3763 . . . . 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 15881 . . 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 15017 . 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 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   0gc0g 13416  SubGrpcsubg 14631  Cntzccntz 14807   Abelcabel 15106   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871
This theorem is referenced by:  baerlem3lem1  32519  baerlem5alem1  32520  baerlem5blem1  32521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872
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