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Theorem lindsmm 27174
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 491 . . . 4  |-  ( F 
C_  B  ->  (
(  _I  |`  F ) LIndF 
S  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
213ad2ant3 980 . . 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 5680 . . . . . 6  |-  (  _I  |`  F ) : F -1-1-onto-> F
4 f1of 5641 . . . . . 6  |-  ( (  _I  |`  F ) : F -1-1-onto-> F  ->  (  _I  |`  F ) : F --> F )
53, 4ax-mp 8 . . . . 5  |-  (  _I  |`  F ) : F --> F
6 simp3 959 . . . . 5  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  F  C_  B )
7 fss 5566 . . . . 5  |-  ( ( (  _I  |`  F ) : F --> F  /\  F  C_  B )  -> 
(  _I  |`  F ) : F --> B )
85, 6, 7sylancr 645 . . . 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 27173 . . . 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 1229 . . 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 247 . 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 16072 . . . 4  |-  ( G  e.  ( S LMHom  T
)  ->  S  e.  LMod )
15143ad2ant1 978 . . 3  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  S  e.  LMod )
169islinds 27155 . . 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 16071 . . . . . . 7  |-  ( G  e.  ( S LMHom  T
)  ->  T  e.  LMod )
19183ad2ant1 978 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  T  e.  LMod )
2019adantr 452 . . . . 5  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G " F )  e.  (LIndS `  T
) )  ->  T  e.  LMod )
21 simpr 448 . . . . 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 5656 . . . . . . . 8  |-  ( ( G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-onto-> ( G " F ) )
23 f1of1 5640 . . . . . . . 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 975 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
2625adantr 452 . . . . 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 27171 . . . . 5  |-  ( ( T  e.  LMod  /\  ( G " F )  e.  (LIndS `  T )  /\  ( G  |`  F ) : F -1-1-> ( G
" F ) )  ->  ( G  |`  F ) LIndF  T )
2820, 21, 26, 27syl3anc 1184 . . . 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 27167 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( G  |`  F ) LIndF  T
)  ->  ran  ( G  |`  F )  e.  (LIndS `  T ) )
3119, 30sylan 458 . . . . 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 2496 . . . 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 806 . . 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 5354 . . . 4  |-  ( G  o.  (  _I  |`  F ) )  =  ( G  |`  F )
3534breq1i 4187 . . 3  |-  ( ( G  o.  (  _I  |`  F ) ) LIndF  T  <->  ( G  |`  F ) LIndF  T )
3633, 35syl6bbr 255 . 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 277 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3288   class class class wbr 4180    _I cid 4461   ran crn 4846    |` cres 4847   "cima 4848    o. ccom 4849   -->wf 5417   -1-1->wf1 5418   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   Basecbs 13432   LModclmod 15913   LMHom clmhm 16058   LIndF clindf 27150  LIndSclinds 27151
This theorem is referenced by:  lindsmm2  27175
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-mnd 14653  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-ghm 14967  df-mgp 15612  df-rng 15626  df-ur 15628  df-lmod 15915  df-lss 15972  df-lsp 16011  df-lmhm 16061  df-lindf 27152  df-linds 27153
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