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Theorem lmimlbs 26976
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 16064 . . . 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 2388 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
4 eqid 2388 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
53, 4lmimf1o 16063 . . . . 5  |-  ( F  e.  ( S LMIso  T
)  ->  F :
( Base `  S ) -1-1-onto-> ( Base `  T ) )
6 f1of1 5614 . . . . 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 26973 . . . . 5  |-  J  C_  (LIndS `  S )
1110sseli 3288 . . . 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 26969 . . 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 2388 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
163, 9, 15lbssp 16079 . . . . 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 5144 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( F " ( Base `  S ) ) )
193, 9lbsss 16077 . . . 4  |-  ( B  e.  J  ->  B  C_  ( Base `  S
) )
20 eqid 2388 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
213, 15, 20lmhmlsp 16053 . . . 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 5622 . . . 4  |-  ( F : ( Base `  S
)
-1-1-onto-> ( Base `  T )  ->  F : ( Base `  S ) -onto-> ( Base `  T ) )
25 foima 5599 . . . 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 2428 . 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 26972 . 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 3264   "cima 4822   -1-1->wf1 5392   -onto->wfo 5393   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   Basecbs 13397   LSpanclspn 15975   LMHom clmhm 16023   LMIso clmim 16024  LBasisclbs 16074  LIndSclinds 26945
This theorem is referenced by:  lmiclbs  26977
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-ghm 14932  df-mgp 15577  df-rng 15591  df-ur 15593  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lmhm 16026  df-lmim 16027  df-lbs 16075  df-lindf 26946  df-linds 26947
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