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Theorem lindsmm 19030
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 502 . . . 4  |-  ( F 
C_  B  ->  (
(  _I  |`  F ) LIndF 
S  <->  ( F  C_  B  /\  (  _I  |`  F ) LIndF 
S ) ) )
213ad2ant3 1017 . . 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 5833 . . . . . 6  |-  (  _I  |`  F ) : F -1-1-onto-> F
4 f1of 5798 . . . . . 6  |-  ( (  _I  |`  F ) : F -1-1-onto-> F  ->  (  _I  |`  F ) : F --> F )
53, 4ax-mp 5 . . . . 5  |-  (  _I  |`  F ) : F --> F
6 simp3 996 . . . . 5  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  F  C_  B )
7 fss 5721 . . . . 5  |-  ( ( (  _I  |`  F ) : F --> F  /\  F  C_  B )  -> 
(  _I  |`  F ) : F --> B )
85, 6, 7sylancr 661 . . . 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 19029 . . . 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 1271 . . 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 17874 . . . 4  |-  ( G  e.  ( S LMHom  T
)  ->  S  e.  LMod )
15143ad2ant1 1015 . . 3  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  S  e.  LMod )
169islinds 19011 . . 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 17873 . . . . . . 7  |-  ( G  e.  ( S LMHom  T
)  ->  T  e.  LMod )
19183ad2ant1 1015 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  T  e.  LMod )
2019adantr 463 . . . . 5  |-  ( ( ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  /\  ( G " F )  e.  (LIndS `  T
) )  ->  T  e.  LMod )
21 simpr 459 . . . . 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 5812 . . . . . . . 8  |-  ( ( G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-onto-> ( G " F ) )
23 f1of1 5797 . . . . . . . 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 1012 . . . . . 6  |-  ( ( G  e.  ( S LMHom 
T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( G  |`  F ) : F -1-1-> ( G " F ) )
2625adantr 463 . . . . 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 19027 . . . . 5  |-  ( ( T  e.  LMod  /\  ( G " F )  e.  (LIndS `  T )  /\  ( G  |`  F ) : F -1-1-> ( G
" F ) )  ->  ( G  |`  F ) LIndF  T )
2820, 21, 26, 27syl3anc 1226 . . . 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 5001 . . . . 5  |-  ( G
" F )  =  ran  ( G  |`  F )
30 lindfrn 19023 . . . . . 6  |-  ( ( T  e.  LMod  /\  ( G  |`  F ) LIndF  T
)  ->  ran  ( G  |`  F )  e.  (LIndS `  T ) )
3119, 30sylan 469 . . . . 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 2546 . . . 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 830 . . 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 5508 . . . 4  |-  ( G  o.  (  _I  |`  F ) )  =  ( G  |`  F )
3534breq1i 4446 . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   class class class wbr 4439    _I cid 4779   ran crn 4989    |` cres 4990   "cima 4991    o. ccom 4992   -->wf 5566   -1-1->wf1 5567   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Basecbs 14716   LModclmod 17707   LMHom clmhm 17860   LIndF clindf 19006  LIndSclinds 19007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-ghm 16464  df-mgp 17337  df-ur 17349  df-ring 17395  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lmhm 17863  df-lindf 19008  df-linds 19009
This theorem is referenced by:  lindsmm2  19031
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