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Theorem lindssnlvec 32035
Description: A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
Assertion
Ref Expression
lindssnlvec  |-  ( ( M  e.  LVec  /\  S  e.  ( Base `  M
)  /\  S  =/=  ( 0g `  M ) )  ->  { S } linIndS  M )

Proof of Theorem lindssnlvec
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 eldifsni 4146 . . . . 5  |-  ( s  e.  ( ( Base `  (Scalar `  M )
)  \  { ( 0g `  (Scalar `  M
) ) } )  ->  s  =/=  ( 0g `  (Scalar `  M
) ) )
21adantl 466 . . . 4  |-  ( ( ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M
) )  /\  s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) )  -> 
s  =/=  ( 0g
`  (Scalar `  M )
) )
3 simpl3 996 . . . 4  |-  ( ( ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M
) )  /\  s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) )  ->  S  =/=  ( 0g `  M ) )
4 eqid 2460 . . . . 5  |-  ( Base `  M )  =  (
Base `  M )
5 eqid 2460 . . . . 5  |-  ( .s
`  M )  =  ( .s `  M
)
6 eqid 2460 . . . . 5  |-  (Scalar `  M )  =  (Scalar `  M )
7 eqid 2460 . . . . 5  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
8 eqid 2460 . . . . 5  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
9 eqid 2460 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
10 simpl1 994 . . . . 5  |-  ( ( ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M
) )  /\  s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) )  ->  M  e.  LVec )
11 eldifi 3619 . . . . . 6  |-  ( s  e.  ( ( Base `  (Scalar `  M )
)  \  { ( 0g `  (Scalar `  M
) ) } )  ->  s  e.  (
Base `  (Scalar `  M
) ) )
1211adantl 466 . . . . 5  |-  ( ( ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M
) )  /\  s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) )  -> 
s  e.  ( Base `  (Scalar `  M )
) )
13 simpl2 995 . . . . 5  |-  ( ( ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M
) )  /\  s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) )  ->  S  e.  ( Base `  M ) )
144, 5, 6, 7, 8, 9, 10, 12, 13lvecvsn0 17531 . . . 4  |-  ( ( ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M
) )  /\  s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) )  -> 
( ( s ( .s `  M ) S )  =/=  ( 0g `  M )  <->  ( s  =/=  ( 0g `  (Scalar `  M ) )  /\  S  =/=  ( 0g `  M ) ) ) )
152, 3, 14mpbir2and 915 . . 3  |-  ( ( ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M
) )  /\  s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) )  -> 
( s ( .s
`  M ) S )  =/=  ( 0g
`  M ) )
1615ralrimiva 2871 . 2  |-  ( ( M  e.  LVec  /\  S  e.  ( Base `  M
)  /\  S  =/=  ( 0g `  M ) )  ->  A. s  e.  ( ( Base `  (Scalar `  M ) )  \  { ( 0g `  (Scalar `  M ) ) } ) ( s ( .s `  M
) S )  =/=  ( 0g `  M
) )
17 lveclmod 17528 . . . . 5  |-  ( M  e.  LVec  ->  M  e. 
LMod )
1817anim1i 568 . . . 4  |-  ( ( M  e.  LVec  /\  S  e.  ( Base `  M
) )  ->  ( M  e.  LMod  /\  S  e.  ( Base `  M
) ) )
19183adant3 1011 . . 3  |-  ( ( M  e.  LVec  /\  S  e.  ( Base `  M
)  /\  S  =/=  ( 0g `  M ) )  ->  ( M  e.  LMod  /\  S  e.  ( Base `  M )
) )
204, 6, 7, 8, 9, 5snlindsntor 32020 . . 3  |-  ( ( M  e.  LMod  /\  S  e.  ( Base `  M
) )  ->  ( A. s  e.  (
( Base `  (Scalar `  M
) )  \  {
( 0g `  (Scalar `  M ) ) } ) ( s ( .s `  M ) S )  =/=  ( 0g `  M )  <->  { S } linIndS  M ) )
2119, 20syl 16 . 2  |-  ( ( M  e.  LVec  /\  S  e.  ( Base `  M
)  /\  S  =/=  ( 0g `  M ) )  ->  ( A. s  e.  ( ( Base `  (Scalar `  M
) )  \  {
( 0g `  (Scalar `  M ) ) } ) ( s ( .s `  M ) S )  =/=  ( 0g `  M )  <->  { S } linIndS  M ) )
2216, 21mpbid 210 1  |-  ( ( M  e.  LVec  /\  S  e.  ( Base `  M
)  /\  S  =/=  ( 0g `  M ) )  ->  { S } linIndS  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    e. wcel 1762    =/= wne 2655   A.wral 2807    \ cdif 3466   {csn 4020   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479  Scalarcsca 14547   .scvsca 14548   0gc0g 14684   LModclmod 17288   LVecclvec 17524   linIndS clininds 31989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-0g 14686  df-gsum 14687  df-mnd 15721  df-grp 15851  df-minusg 15852  df-mulg 15854  df-cntz 16143  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-drng 17174  df-lmod 17290  df-lvec 17525  df-linc 31955  df-lininds 31991
This theorem is referenced by: (None)
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