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Theorem lindff1 27258
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 27253 . . 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 5208 . . . . . . . . . 10  |-  ( F
" ( dom  F  \  { y } ) )  C_  ran  F
8 frn 5589 . . . . . . . . . . 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 3351 . . . . . . . . 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 2435 . . . . . . . . 9  |-  ( LSpan `  W )  =  (
LSpan `  W )
133, 12lspssid 16053 . . . . . . . 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 5585 . . . . . . . . . . 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 3919 . . . . . . . . . . 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 5965 . . . . . . . 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 3341 . . . . . 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 27256 . . . . . . 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 2688 . . . . . 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 2661 . . 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 2790 . 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 5996 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    \ cdif 3309    C_ wss 3312   {csn 3806   class class class wbr 4204   dom cdm 4870   ran crn 4871   "cima 4873   Fun wfun 5440   -->wf 5442   -1-1->wf1 5443   ` cfv 5446   Basecbs 13461  Scalarcsca 13524   LModclmod 15942   LSpanclspn 16039  NzRingcnzr 16320   LIndF clindf 27242
This theorem is referenced by:  islindf3  27264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lss 16001  df-lsp 16040  df-nzr 16321  df-lindf 27244
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