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Theorem lmimlbs 26638
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 15744 . . . 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 2256 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
4 eqid 2256 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
53, 4lmimf1o 15743 . . . . 5  |-  ( F  e.  ( S LMIso  T
)  ->  F :
( Base `  S ) -1-1-onto-> ( Base `  T ) )
6 f1of1 5374 . . . . 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 26635 . . . . 5  |-  J  C_  (LIndS `  S )
1110sseli 3118 . . . 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 26631 . . 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 2256 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
163, 9, 15lbssp 15759 . . . . 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 4965 . . 3  |-  ( ( F  e.  ( S LMIso 
T )  /\  B  e.  J )  ->  ( F " ( ( LSpan `  S ) `  B
) )  =  ( F " ( Base `  S ) ) )
193, 9lbsss 15757 . . . 4  |-  ( B  e.  J  ->  B  C_  ( Base `  S
) )
20 eqid 2256 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
213, 15, 20lmhmlsp 15733 . . . 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 5382 . . . 4  |-  ( F : ( Base `  S
)
-1-1-onto-> ( Base `  T )  ->  F : ( Base `  S ) -onto-> ( Base `  T ) )
25 foima 5359 . . . 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 2296 . 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 26634 . 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 3094   "cima 4629   -1-1->wf1 4635   -onto->wfo 4636   -1-1-onto->wf1o 4637   ` cfv 4638  (class class class)co 5757   Basecbs 13075   LSpanclspn 15655   LMHom clmhm 15703   LMIso clmim 15704  LBasisclbs 15754  LIndSclinds 26607
This theorem is referenced by:  lmiclbs  26639
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-0g 13331  df-mnd 14294  df-grp 14416  df-minusg 14417  df-sbg 14418  df-subg 14545  df-ghm 14608  df-mgp 15253  df-ring 15267  df-ur 15269  df-lmod 15556  df-lss 15617  df-lsp 15656  df-lmhm 15706  df-lmim 15707  df-lbs 15755  df-lindf 26608  df-linds 26609
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