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Theorem lindsmm 27298
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 5511 . . . . . 6  |-  (  _I  |`  F ) : F -1-1-onto-> F
4 f1of 5472 . . . . . 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 5397 . . . . 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 27297 . . . 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 15790 . . . 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 27279 . . 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 15789 . . . . . . 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 5487 . . . . . . . 8  |-  ( ( G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-onto-> ( G " F ) )
23 f1of1 5471 . . . . . . . 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 27295 . . . . 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 4702 . . . . 5  |-  ( G
" F )  =  ran  ( G  |`  F )
30 lindfrn 27291 . . . . . 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 2367 . . . 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 5190 . . . 4  |-  ( G  o.  (  _I  |`  F ) )  =  ( G  |`  F )
3534breq1i 4030 . . 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 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023    _I cid 4304   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148   LModclmod 15627   LMHom clmhm 15776   LIndF clindf 27274  LIndSclinds 27275
This theorem is referenced by:  lindsmm2  27299
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-1st 6122  df-2nd 6123  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-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lindf 27276  df-linds 27277
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