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Theorem lindsmm 18262
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 504 . . . 4  |-  ( F 
C_  B  ->  (
(  _I  |`  F ) LIndF 
S  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
213ad2ant3 1011 . . 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 5681 . . . . . 6  |-  (  _I  |`  F ) : F -1-1-onto-> F
4 f1of 5646 . . . . . 6  |-  ( (  _I  |`  F ) : F -1-1-onto-> F  ->  (  _I  |`  F ) : F --> F )
53, 4ax-mp 5 . . . . 5  |-  (  _I  |`  F ) : F --> F
6 simp3 990 . . . . 5  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  F  C_  B )
7 fss 5572 . . . . 5  |-  ( ( (  _I  |`  F ) : F --> F  /\  F  C_  B )  -> 
(  _I  |`  F ) : F --> B )
85, 6, 7sylancr 663 . . . 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 18261 . . . 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 1263 . . 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 255 . 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 17119 . . . 4  |-  ( G  e.  ( S LMHom  T
)  ->  S  e.  LMod )
15143ad2ant1 1009 . . 3  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  S  e.  LMod )
169islinds 18243 . . 3  |-  ( S  e.  LMod  ->  ( F  e.  (LIndS `  S
)  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
1715, 16syl 16 . 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 17118 . . . . . . 7  |-  ( G  e.  ( S LMHom  T
)  ->  T  e.  LMod )
19183ad2ant1 1009 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  T  e.  LMod )
2019adantr 465 . . . . 5  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G " F )  e.  (LIndS `  T
) )  ->  T  e.  LMod )
21 simpr 461 . . . . 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 5660 . . . . . . . 8  |-  ( ( G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-onto-> ( G " F ) )
23 f1of1 5645 . . . . . . . 8  |-  ( ( G  |`  F ) : F -1-1-onto-> ( G " F
)  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
2422, 23syl 16 . . . . . . 7  |-  ( ( G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
25243adant1 1006 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
2625adantr 465 . . . . 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 18259 . . . . 5  |-  ( ( T  e.  LMod  /\  ( G " F )  e.  (LIndS `  T )  /\  ( G  |`  F ) : F -1-1-> ( G
" F ) )  ->  ( G  |`  F ) LIndF  T )
2820, 21, 26, 27syl3anc 1218 . . . 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 4858 . . . . 5  |-  ( G
" F )  =  ran  ( G  |`  F )
30 lindfrn 18255 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( G  |`  F ) LIndF  T
)  ->  ran  ( G  |`  F )  e.  (LIndS `  T ) )
3119, 30sylan 471 . . . . 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 2527 . . . 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 828 . . 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 5360 . . . 4  |-  ( G  o.  (  _I  |`  F ) )  =  ( G  |`  F )
3534breq1i 4304 . . 3  |-  ( ( G  o.  (  _I  |`  F ) ) LIndF  T  <->  ( G  |`  F ) LIndF  T )
3633, 35syl6bbr 263 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3333   class class class wbr 4297    _I cid 4636   ran crn 4846    |` cres 4847   "cima 4848    o. ccom 4849   -->wf 5419   -1-1->wf1 5420   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   Basecbs 14179   LModclmod 16953   LMHom clmhm 17105   LIndF clindf 18238  LIndSclinds 18239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-ghm 15750  df-mgp 16597  df-ur 16609  df-rng 16652  df-lmod 16955  df-lss 17019  df-lsp 17058  df-lmhm 17108  df-lindf 18240  df-linds 18241
This theorem is referenced by:  lindsmm2  18263
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