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Theorem lindff1 27290
Description: A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
lindff1.b  |-  B  =  ( Base `  W
)
lindff1.l  |-  L  =  (Scalar `  W )
Assertion
Ref Expression
lindff1  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F : dom  F -1-1-> B )

Proof of Theorem lindff1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . 3  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F LIndF  W )
2 simp1 955 . . 3  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  W  e.  LMod )
3 lindff1.b . . . 4  |-  B  =  ( Base `  W
)
43lindff 27285 . . 3  |-  ( ( F LIndF  W  /\  W  e.  LMod )  ->  F : dom  F --> B )
51, 2, 4syl2anc 642 . 2  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F : dom  F --> B )
6 simpl1 958 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  W  e.  LMod )
7 imassrn 5025 . . . . . . . . . 10  |-  ( F
" ( dom  F  \  { y } ) )  C_  ran  F
8 frn 5395 . . . . . . . . . . 11  |-  ( F : dom  F --> B  ->  ran  F  C_  B )
95, 8syl 15 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  ran  F  C_  B
)
107, 9syl5ss 3190 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  ( F " ( dom  F  \  { y } ) )  C_  B )
1110adantr 451 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  -> 
( F " ( dom  F  \  { y } ) )  C_  B )
12 eqid 2283 . . . . . . . . 9  |-  ( LSpan `  W )  =  (
LSpan `  W )
133, 12lspssid 15742 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( F " ( dom  F  \  { y } ) )  C_  B )  ->  ( F " ( dom  F  \  { y } ) )  C_  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { y } ) ) ) )
146, 11, 13syl2anc 642 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  -> 
( F " ( dom  F  \  { y } ) )  C_  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { y } ) ) ) )
15 ffun 5391 . . . . . . . . . . 11  |-  ( F : dom  F --> B  ->  Fun  F )
165, 15syl 15 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  Fun  F )
1716adantr 451 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  Fun  F )
18 simprll 738 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  x  e.  dom  F )
1917, 18jca 518 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  -> 
( Fun  F  /\  x  e.  dom  F ) )
20 eldifsn 3749 . . . . . . . . . . 11  |-  ( x  e.  ( dom  F  \  { y } )  <-> 
( x  e.  dom  F  /\  x  =/=  y
) )
2120biimpri 197 . . . . . . . . . 10  |-  ( ( x  e.  dom  F  /\  x  =/=  y
)  ->  x  e.  ( dom  F  \  {
y } ) )
2221adantlr 695 . . . . . . . . 9  |-  ( ( ( x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y )  ->  x  e.  ( dom  F  \  { y } ) )
2322adantl 452 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  x  e.  ( dom  F 
\  { y } ) )
24 funfvima 5753 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( x  e.  ( dom  F  \  {
y } )  -> 
( F `  x
)  e.  ( F
" ( dom  F  \  { y } ) ) ) )
2519, 23, 24sylc 56 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  -> 
( F `  x
)  e.  ( F
" ( dom  F  \  { y } ) ) )
2614, 25sseldd 3181 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  -> 
( F `  x
)  e.  ( (
LSpan `  W ) `  ( F " ( dom 
F  \  { y } ) ) ) )
27 simpl2 959 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  L  e. NzRing )
28 simpl3 960 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  F LIndF  W )
29 simprlr 739 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  -> 
y  e.  dom  F
)
30 lindff1.l . . . . . . . 8  |-  L  =  (Scalar `  W )
3112, 30lindfind2 27288 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing )  /\  F LIndF  W  /\  y  e.  dom  F )  ->  -.  ( F `  y )  e.  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { y } ) ) ) )
326, 27, 28, 29, 31syl211anc 1188 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  -.  ( F `  y
)  e.  ( (
LSpan `  W ) `  ( F " ( dom 
F  \  { y } ) ) ) )
33 nelne2 2536 . . . . . 6  |-  ( ( ( F `  x
)  e.  ( (
LSpan `  W ) `  ( F " ( dom 
F  \  { y } ) ) )  /\  -.  ( F `
 y )  e.  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { y } ) ) ) )  -> 
( F `  x
)  =/=  ( F `
 y ) )
3426, 32, 33syl2anc 642 . . . . 5  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  -> 
( F `  x
)  =/=  ( F `
 y ) )
3534expr 598 . . . 4  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( x  e.  dom  F  /\  y  e.  dom  F ) )  ->  ( x  =/=  y  ->  ( F `  x )  =/=  ( F `  y )
) )
3635necon4d 2509 . . 3  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( x  e.  dom  F  /\  y  e.  dom  F ) )  ->  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) )
3736ralrimivva 2635 . 2  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  A. x  e.  dom  F A. y  e.  dom  F ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
38 dff13 5783 . 2  |-  ( F : dom  F -1-1-> B  <->  ( F : dom  F --> B  /\  A. x  e. 
dom  F A. y  e.  dom  F ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
395, 37, 38sylanbrc 645 1  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F : dom  F -1-1-> B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252   ` cfv 5255   Basecbs 13148  Scalarcsca 13211   LModclmod 15627   LSpanclspn 15728  NzRingcnzr 16009   LIndF clindf 27274
This theorem is referenced by:  islindf3  27296
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-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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729  df-nzr 16010  df-lindf 27276
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