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Theorem lindff1 26952
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 959 . . 3  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F LIndF  W )
2 simp1 957 . . 3  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  W  e.  LMod )
3 lindff1.b . . . 4  |-  B  =  ( Base `  W
)
43lindff 26947 . . 3  |-  ( ( F LIndF  W  /\  W  e.  LMod )  ->  F : dom  F --> B )
51, 2, 4syl2anc 643 . 2  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F : dom  F --> B )
6 simpl1 960 . . . . . . . 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 5149 . . . . . . . . . 10  |-  ( F
" ( dom  F  \  { y } ) )  C_  ran  F
8 frn 5530 . . . . . . . . . . 11  |-  ( F : dom  F --> B  ->  ran  F  C_  B )
95, 8syl 16 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  ran  F  C_  B
)
107, 9syl5ss 3295 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  ( F " ( dom  F  \  { y } ) )  C_  B )
1110adantr 452 . . . . . . . 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 2380 . . . . . . . . 9  |-  ( LSpan `  W )  =  (
LSpan `  W )
133, 12lspssid 15981 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( F " ( dom  F  \  { y } ) )  C_  B )  ->  ( F " ( dom  F  \  { y } ) )  C_  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { y } ) ) ) )
146, 11, 13syl2anc 643 . . . . . . 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 5526 . . . . . . . . . . 11  |-  ( F : dom  F --> B  ->  Fun  F )
165, 15syl 16 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  Fun  F )
1716adantr 452 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W
)  /\  ( (
x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y ) )  ->  Fun  F )
18 simprll 739 . . . . . . . . 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 519 . . . . . . . 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 3863 . . . . . . . . . . 11  |-  ( x  e.  ( dom  F  \  { y } )  <-> 
( x  e.  dom  F  /\  x  =/=  y
) )
2120biimpri 198 . . . . . . . . . 10  |-  ( ( x  e.  dom  F  /\  x  =/=  y
)  ->  x  e.  ( dom  F  \  {
y } ) )
2221adantlr 696 . . . . . . . . 9  |-  ( ( ( x  e.  dom  F  /\  y  e.  dom  F )  /\  x  =/=  y )  ->  x  e.  ( dom  F  \  { y } ) )
2322adantl 453 . . . . . . . 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 5905 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( x  e.  ( dom  F  \  {
y } )  -> 
( F `  x
)  e.  ( F
" ( dom  F  \  { y } ) ) ) )
2519, 23, 24sylc 58 . . . . . . 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 3285 . . . . . 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 961 . . . . . . 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 962 . . . . . . 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 740 . . . . . . 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 26950 . . . . . . 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 1190 . . . . . 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 2633 . . . . . 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 643 . . . . 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 599 . . . 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 2606 . . 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 2734 . 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 5936 . 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 646 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642    \ cdif 3253    C_ wss 3256   {csn 3750   class class class wbr 4146   dom cdm 4811   ran crn 4812   "cima 4814   Fun wfun 5381   -->wf 5383   -1-1->wf1 5384   ` cfv 5387   Basecbs 13389  Scalarcsca 13452   LModclmod 15870   LSpanclspn 15967  NzRingcnzr 16248   LIndF clindf 26936
This theorem is referenced by:  islindf3  26958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-0g 13647  df-mnd 14610  df-grp 14732  df-mgp 15569  df-rng 15583  df-ur 15585  df-lmod 15872  df-lss 15929  df-lsp 15968  df-nzr 16249  df-lindf 26938
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