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Theorem lmimlbs 26977
Description: The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Hypotheses
Ref Expression
lmimlbs.j  |-  J  =  (LBasis `  S )
lmimlbs.k  |-  K  =  (LBasis `  T )
Assertion
Ref Expression
lmimlbs  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " B )  e.  K )

Proof of Theorem lmimlbs
StepHypRef Expression
1 lmimlmhm 16065 . . . 4  |-  ( F  e.  ( S LMIso  T
)  ->  F  e.  ( S LMHom  T ) )
21adantr 452 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F  e.  ( S LMHom  T ) )
3 eqid 2389 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
4 eqid 2389 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
53, 4lmimf1o 16064 . . . . 5  |-  ( F  e.  ( S LMIso  T
)  ->  F :
( Base `  S ) -1-1-onto-> ( Base `  T ) )
6 f1of1 5615 . . . . 5  |-  ( F : ( Base `  S
)
-1-1-onto-> ( Base `  T )  ->  F : ( Base `  S ) -1-1-> ( Base `  T ) )
75, 6syl 16 . . . 4  |-  ( F  e.  ( S LMIso  T
)  ->  F :
( Base `  S ) -1-1-> ( Base `  T
) )
87adantr 452 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F : ( Base `  S
) -1-1-> ( Base `  T
) )
9 lmimlbs.j . . . . . 6  |-  J  =  (LBasis `  S )
109lbslinds 26974 . . . . 5  |-  J  C_  (LIndS `  S )
1110sseli 3289 . . . 4  |-  ( B  e.  J  ->  B  e.  (LIndS `  S )
)
1211adantl 453 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  B  e.  (LIndS `  S )
)
133, 4lindsmm2 26970 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  F : ( Base `  S
) -1-1-> ( Base `  T
)  /\  B  e.  (LIndS `  S ) )  ->  ( F " B )  e.  (LIndS `  T ) )
142, 8, 12, 13syl3anc 1184 . 2  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " B )  e.  (LIndS `  T )
)
15 eqid 2389 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
163, 9, 15lbssp 16080 . . . . 5  |-  ( B  e.  J  ->  (
( LSpan `  S ) `  B )  =  (
Base `  S )
)
1716adantl 453 . . . 4  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  (
( LSpan `  S ) `  B )  =  (
Base `  S )
)
1817imaeq2d 5145 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( F " ( Base `  S ) ) )
193, 9lbsss 16078 . . . 4  |-  ( B  e.  J  ->  B  C_  ( Base `  S
) )
20 eqid 2389 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
213, 15, 20lmhmlsp 16054 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  B  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( ( LSpan `  T ) `  ( F " B
) ) )
221, 19, 21syl2an 464 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( ( LSpan `  T ) `  ( F " B
) ) )
235adantr 452 . . . 4  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F : ( Base `  S
)
-1-1-onto-> ( Base `  T )
)
24 f1ofo 5623 . . . 4  |-  ( F : ( Base `  S
)
-1-1-onto-> ( Base `  T )  ->  F : ( Base `  S ) -onto-> ( Base `  T ) )
25 foima 5600 . . . 4  |-  ( F : ( Base `  S
) -onto-> ( Base `  T
)  ->  ( F " ( Base `  S
) )  =  (
Base `  T )
)
2623, 24, 253syl 19 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( Base `  S
) )  =  (
Base `  T )
)
2718, 22, 263eqtr3d 2429 . 2  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  (
( LSpan `  T ) `  ( F " B
) )  =  (
Base `  T )
)
28 lmimlbs.k . . 3  |-  K  =  (LBasis `  T )
294, 28, 20islbs4 26973 . 2  |-  ( ( F " B )  e.  K  <->  ( ( F " B )  e.  (LIndS `  T )  /\  ( ( LSpan `  T
) `  ( F " B ) )  =  ( Base `  T
) ) )
3014, 27, 29sylanbrc 646 1  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " B )  e.  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   "cima 4823   -1-1->wf1 5393   -onto->wfo 5394   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   Basecbs 13398   LSpanclspn 15976   LMHom clmhm 16024   LMIso clmim 16025  LBasisclbs 16075  LIndSclinds 26946
This theorem is referenced by:  lmiclbs  26978
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-ghm 14933  df-mgp 15578  df-rng 15592  df-ur 15594  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lmhm 16027  df-lmim 16028  df-lbs 16076  df-lindf 26947  df-linds 26948
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