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Theorem lmimlbs 18263
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 17143 . . . 4  |-  ( F  e.  ( S LMIso  T
)  ->  F  e.  ( S LMHom  T ) )
21adantr 465 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F  e.  ( S LMHom  T ) )
3 eqid 2441 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
4 eqid 2441 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
53, 4lmimf1o 17142 . . . . 5  |-  ( F  e.  ( S LMIso  T
)  ->  F :
( Base `  S ) -1-1-onto-> ( Base `  T ) )
6 f1of1 5638 . . . . 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 465 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F : ( Base `  S
) -1-1-> ( Base `  T
) )
9 lmimlbs.j . . . . . 6  |-  J  =  (LBasis `  S )
109lbslinds 18260 . . . . 5  |-  J  C_  (LIndS `  S )
1110sseli 3350 . . . 4  |-  ( B  e.  J  ->  B  e.  (LIndS `  S )
)
1211adantl 466 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  B  e.  (LIndS `  S )
)
133, 4lindsmm2 18256 . . 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 1218 . 2  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " B )  e.  (LIndS `  T )
)
15 eqid 2441 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
163, 9, 15lbssp 17158 . . . . 5  |-  ( B  e.  J  ->  (
( LSpan `  S ) `  B )  =  (
Base `  S )
)
1716adantl 466 . . . 4  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  (
( LSpan `  S ) `  B )  =  (
Base `  S )
)
1817imaeq2d 5167 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( F " ( Base `  S ) ) )
193, 9lbsss 17156 . . . 4  |-  ( B  e.  J  ->  B  C_  ( Base `  S
) )
20 eqid 2441 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
213, 15, 20lmhmlsp 17128 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  B  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( ( LSpan `  T ) `  ( F " B
) ) )
221, 19, 21syl2an 477 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( ( LSpan `  T ) `  ( F " B
) ) )
235adantr 465 . . . 4  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F : ( Base `  S
)
-1-1-onto-> ( Base `  T )
)
24 f1ofo 5646 . . . 4  |-  ( F : ( Base `  S
)
-1-1-onto-> ( Base `  T )  ->  F : ( Base `  S ) -onto-> ( Base `  T ) )
25 foima 5623 . . . 4  |-  ( F : ( Base `  S
) -onto-> ( Base `  T
)  ->  ( F " ( Base `  S
) )  =  (
Base `  T )
)
2623, 24, 253syl 20 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( Base `  S
) )  =  (
Base `  T )
)
2718, 22, 263eqtr3d 2481 . 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 18259 . 2  |-  ( ( F " B )  e.  K  <->  ( ( F " B )  e.  (LIndS `  T )  /\  ( ( LSpan `  T
) `  ( F " B ) )  =  ( Base `  T
) ) )
3014, 27, 29sylanbrc 664 1  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " B )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3326   "cima 4841   -1-1->wf1 5413   -onto->wfo 5414   -1-1-onto->wf1o 5415   ` cfv 5416  (class class class)co 6089   Basecbs 14172   LSpanclspn 17050   LMHom clmhm 17098   LMIso clmim 17099  LBasisclbs 17153  LIndSclinds 18232
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545  df-subg 15676  df-ghm 15743  df-mgp 16590  df-ur 16602  df-rng 16645  df-lmod 16948  df-lss 17012  df-lsp 17051  df-lmhm 17101  df-lmim 17102  df-lbs 17154  df-lindf 18233  df-linds 18234
This theorem is referenced by:  lmiclbs  18264
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