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Theorem lmimlbs 26472
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 15652 . . . 4  |-  ( F  e.  ( S LMIso  T
)  ->  F  e.  ( S LMHom  T ) )
21adantr 453 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F  e.  ( S LMHom  T ) )
3 eqid 2253 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
4 eqid 2253 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
53, 4lmimf1o 15651 . . . . 5  |-  ( F  e.  ( S LMIso  T
)  ->  F :
( Base `  S ) -1-1-onto-> ( Base `  T ) )
6 f1of1 5328 . . . . 5  |-  ( F : ( Base `  S
)
-1-1-onto-> ( Base `  T )  ->  F : ( Base `  S ) -1-1-> ( Base `  T ) )
75, 6syl 17 . . . 4  |-  ( F  e.  ( S LMIso  T
)  ->  F :
( Base `  S ) -1-1-> ( Base `  T
) )
87adantr 453 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F : ( Base `  S
) -1-1-> ( Base `  T
) )
9 lmimlbs.j . . . . . 6  |-  J  =  (LBasis `  S )
109lbslinds 26469 . . . . 5  |-  J  C_  (LIndS `  S )
1110sseli 3099 . . . 4  |-  ( B  e.  J  ->  B  e.  (LIndS `  S )
)
1211adantl 454 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  B  e.  (LIndS `  S )
)
133, 4lindsmm2 26465 . . 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 1187 . 2  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " B )  e.  (LIndS `  T )
)
15 eqid 2253 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
163, 9, 15lbssp 15667 . . . . 5  |-  ( B  e.  J  ->  (
( LSpan `  S ) `  B )  =  (
Base `  S )
)
1716adantl 454 . . . 4  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  (
( LSpan `  S ) `  B )  =  (
Base `  S )
)
1817imaeq2d 4919 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( F " ( Base `  S ) ) )
193, 9lbsss 15665 . . . 4  |-  ( B  e.  J  ->  B  C_  ( Base `  S
) )
20 eqid 2253 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
213, 15, 20lmhmlsp 15641 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  B  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( ( LSpan `  T ) `  ( F " B
) ) )
221, 19, 21syl2an 465 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( ( LSpan `  T ) `  ( F " B
) ) )
235adantr 453 . . . 4  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  F : ( Base `  S
)
-1-1-onto-> ( Base `  T )
)
24 f1ofo 5336 . . . 4  |-  ( F : ( Base `  S
)
-1-1-onto-> ( Base `  T )  ->  F : ( Base `  S ) -onto-> ( Base `  T ) )
25 foima 5313 . . . 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 2293 . 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 26468 . 2  |-  ( ( F " B )  e.  K  <->  ( ( F " B )  e.  (LIndS `  T )  /\  ( ( LSpan `  T
) `  ( F " B ) )  =  ( Base `  T
) ) )
3014, 27, 29sylanbrc 648 1  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " B )  e.  K )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3078   "cima 4583   -1-1->wf1 4589   -onto->wfo 4590   -1-1-onto->wf1o 4591   ` cfv 4592  (class class class)co 5710   Basecbs 13022   LSpanclspn 15563   LMHom clmhm 15611   LMIso clmim 15612  LBasisclbs 15662  LIndSclinds 26441
This theorem is referenced by:  lmiclbs  26473
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-0g 13278  df-mnd 14202  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-ghm 14516  df-mgp 15161  df-ring 15175  df-ur 15177  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lmhm 15614  df-lmim 15615  df-lbs 15663  df-lindf 26442  df-linds 26443
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