Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lindsmm Unicode version

Theorem lindsmm 26621
Description: Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Hypotheses
Ref Expression
lindfmm.b  |-  B  =  ( Base `  S
)
lindfmm.c  |-  C  =  ( Base `  T
)
Assertion
Ref Expression
lindsmm  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( F  e.  (LIndS `  S
)  <->  ( G " F )  e.  (LIndS `  T ) ) )

Proof of Theorem lindsmm
StepHypRef Expression
1 ibar 490 . . . 4  |-  ( F 
C_  B  ->  (
(  _I  |`  F ) LIndF 
S  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
213ad2ant3 978 . . 3  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  (
(  _I  |`  F ) LIndF 
S  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
3 f1oi 5591 . . . . . 6  |-  (  _I  |`  F ) : F -1-1-onto-> F
4 f1of 5552 . . . . . 6  |-  ( (  _I  |`  F ) : F -1-1-onto-> F  ->  (  _I  |`  F ) : F --> F )
53, 4ax-mp 8 . . . . 5  |-  (  _I  |`  F ) : F --> F
6 simp3 957 . . . . 5  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  F  C_  B )
7 fss 5477 . . . . 5  |-  ( ( (  _I  |`  F ) : F --> F  /\  F  C_  B )  -> 
(  _I  |`  F ) : F --> B )
85, 6, 7sylancr 644 . . . 4  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  (  _I  |`  F ) : F --> B )
9 lindfmm.b . . . . 5  |-  B  =  ( Base `  S
)
10 lindfmm.c . . . . 5  |-  C  =  ( Base `  T
)
119, 10lindfmm 26620 . . . 4  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  (  _I  |`  F ) : F --> B )  -> 
( (  _I  |`  F ) LIndF 
S  <->  ( G  o.  (  _I  |`  F ) ) LIndF  T ) )
128, 11syld3an3 1227 . . 3  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  (
(  _I  |`  F ) LIndF 
S  <->  ( G  o.  (  _I  |`  F ) ) LIndF  T ) )
132, 12bitr3d 246 . 2  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  (
( F  C_  B  /\  (  _I  |`  F ) LIndF 
S )  <->  ( G  o.  (  _I  |`  F ) ) LIndF  T ) )
14 lmhmlmod1 15883 . . . 4  |-  ( G  e.  ( S LMHom  T
)  ->  S  e.  LMod )
15143ad2ant1 976 . . 3  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  S  e.  LMod )
169islinds 26602 . . 3  |-  ( S  e.  LMod  ->  ( F  e.  (LIndS `  S
)  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
1715, 16syl 15 . 2  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( F  e.  (LIndS `  S
)  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
18 lmhmlmod2 15882 . . . . . . 7  |-  ( G  e.  ( S LMHom  T
)  ->  T  e.  LMod )
19183ad2ant1 976 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  T  e.  LMod )
2019adantr 451 . . . . 5  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G " F )  e.  (LIndS `  T
) )  ->  T  e.  LMod )
21 simpr 447 . . . . 5  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G " F )  e.  (LIndS `  T
) )  ->  ( G " F )  e.  (LIndS `  T )
)
22 f1ores 5567 . . . . . . . 8  |-  ( ( G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-onto-> ( G " F ) )
23 f1of1 5551 . . . . . . . 8  |-  ( ( G  |`  F ) : F -1-1-onto-> ( G " F
)  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
2422, 23syl 15 . . . . . . 7  |-  ( ( G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
25243adant1 973 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
2625adantr 451 . . . . 5  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G " F )  e.  (LIndS `  T
) )  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
27 f1linds 26618 . . . . 5  |-  ( ( T  e.  LMod  /\  ( G " F )  e.  (LIndS `  T )  /\  ( G  |`  F ) : F -1-1-> ( G
" F ) )  ->  ( G  |`  F ) LIndF  T )
2820, 21, 26, 27syl3anc 1182 . . . 4  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G " F )  e.  (LIndS `  T
) )  ->  ( G  |`  F ) LIndF  T
)
29 df-ima 4781 . . . . 5  |-  ( G
" F )  =  ran  ( G  |`  F )
30 lindfrn 26614 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( G  |`  F ) LIndF  T
)  ->  ran  ( G  |`  F )  e.  (LIndS `  T ) )
3119, 30sylan 457 . . . . 5  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G  |`  F ) LIndF 
T )  ->  ran  ( G  |`  F )  e.  (LIndS `  T
) )
3229, 31syl5eqel 2442 . . . 4  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G  |`  F ) LIndF 
T )  ->  ( G " F )  e.  (LIndS `  T )
)
3328, 32impbida 805 . . 3  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  (
( G " F
)  e.  (LIndS `  T )  <->  ( G  |`  F ) LIndF  T ) )
34 coires1 5269 . . . 4  |-  ( G  o.  (  _I  |`  F ) )  =  ( G  |`  F )
3534breq1i 4109 . . 3  |-  ( ( G  o.  (  _I  |`  F ) ) LIndF  T  <->  ( G  |`  F ) LIndF  T )
3633, 35syl6bbr 254 . 2  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  (
( G " F
)  e.  (LIndS `  T )  <->  ( G  o.  (  _I  |`  F ) ) LIndF  T ) )
3713, 17, 363bitr4d 276 1  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( F  e.  (LIndS `  S
)  <->  ( G " F )  e.  (LIndS `  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    C_ wss 3228   class class class wbr 4102    _I cid 4383   ran crn 4769    |` cres 4770   "cima 4771    o. ccom 4772   -->wf 5330   -1-1->wf1 5331   -1-1-onto->wf1o 5333   ` cfv 5334  (class class class)co 5942   Basecbs 13239   LModclmod 15720   LMHom clmhm 15869   LIndF clindf 26597  LIndSclinds 26598
This theorem is referenced by:  lindsmm2  26622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-0g 13497  df-mnd 14460  df-grp 14582  df-minusg 14583  df-sbg 14584  df-subg 14711  df-ghm 14774  df-mgp 15419  df-rng 15433  df-ur 15435  df-lmod 15722  df-lss 15783  df-lsp 15822  df-lmhm 15872  df-lindf 26599  df-linds 26600
  Copyright terms: Public domain W3C validator