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Theorem lmhmpreima 18258
Description: The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmima.x  |-  X  =  ( LSubSp `  S )
lmhmima.y  |-  Y  =  ( LSubSp `  T )
Assertion
Ref Expression
lmhmpreima  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  X )

Proof of Theorem lmhmpreima
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 18241 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
21adantr 466 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  F  e.  ( S  GrpHom  T ) )
3 lmhmlmod2 18242 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
4 lmhmima.y . . . . 5  |-  Y  =  ( LSubSp `  T )
54lsssubg 18167 . . . 4  |-  ( ( T  e.  LMod  /\  U  e.  Y )  ->  U  e.  (SubGrp `  T )
)
63, 5sylan 473 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  U  e.  (SubGrp `  T )
)
7 ghmpreima 16891 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (SubGrp `  T )
)  ->  ( `' F " U )  e.  (SubGrp `  S )
)
82, 6, 7syl2anc 665 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  (SubGrp `  S
) )
9 lmhmlmod1 18243 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
109ad2antrr 730 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  S  e.  LMod )
11 simprl 762 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  a  e.  (
Base `  (Scalar `  S
) ) )
12 cnvimass 5203 . . . . . . . 8  |-  ( `' F " U ) 
C_  dom  F
13 eqid 2422 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2422 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 18244 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
1615adantr 466 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  F : ( Base `  S
) --> ( Base `  T
) )
17 fdm 5746 . . . . . . . . 9  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
1816, 17syl 17 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  dom  F  =  ( Base `  S
) )
1912, 18syl5sseq 3512 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U ) 
C_  ( Base `  S
) )
2019sselda 3464 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  b  e.  ( `' F " U ) )  ->  b  e.  ( Base `  S )
)
2120adantrl 720 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  b  e.  (
Base `  S )
)
22 eqid 2422 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
23 eqid 2422 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
24 eqid 2422 . . . . . 6  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
2513, 22, 23, 24lmodvscl 18095 . . . . 5  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  ( Base `  S ) )  -> 
( a ( .s
`  S ) b )  e.  ( Base `  S ) )
2610, 11, 21, 25syl3anc 1264 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( a ( .s `  S ) b )  e.  (
Base `  S )
)
27 simpll 758 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  F  e.  ( S LMHom  T ) )
28 eqid 2422 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
2922, 24, 13, 23, 28lmhmlin 18245 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  ( Base `  S ) )  -> 
( F `  (
a ( .s `  S ) b ) )  =  ( a ( .s `  T
) ( F `  b ) ) )
3027, 11, 21, 29syl3anc 1264 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( F `  ( a ( .s
`  S ) b ) )  =  ( a ( .s `  T ) ( F `
 b ) ) )
313ad2antrr 730 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  T  e.  LMod )
32 simplr 760 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  U  e.  Y
)
33 eqid 2422 . . . . . . . . . . . 12  |-  (Scalar `  T )  =  (Scalar `  T )
3422, 33lmhmsca 18240 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
3534adantr 466 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (Scalar `  T )  =  (Scalar `  S ) )
3635fveq2d 5881 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
3736eleq2d 2492 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (
a  e.  ( Base `  (Scalar `  T )
)  <->  a  e.  (
Base `  (Scalar `  S
) ) ) )
3837biimpar 487 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  a  e.  ( Base `  (Scalar `  S
) ) )  -> 
a  e.  ( Base `  (Scalar `  T )
) )
3938adantrr 721 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  a  e.  (
Base `  (Scalar `  T
) ) )
40 ffun 5744 . . . . . . . . 9  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Fun  F )
4116, 40syl 17 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  Fun  F )
4241adantr 466 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  Fun  F )
43 simprr 764 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  b  e.  ( `' F " U ) )
44 fvimacnvi 6007 . . . . . . 7  |-  ( ( Fun  F  /\  b  e.  ( `' F " U ) )  -> 
( F `  b
)  e.  U )
4542, 43, 44syl2anc 665 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( F `  b )  e.  U
)
46 eqid 2422 . . . . . . 7  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
4733, 28, 46, 4lssvscl 18165 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  U  e.  Y )  /\  ( a  e.  ( Base `  (Scalar `  T ) )  /\  ( F `  b )  e.  U ) )  ->  ( a ( .s `  T ) ( F `  b
) )  e.  U
)
4831, 32, 39, 45, 47syl22anc 1265 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( a ( .s `  T ) ( F `  b
) )  e.  U
)
4930, 48eqeltrd 2510 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( F `  ( a ( .s
`  S ) b ) )  e.  U
)
50 ffn 5742 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
51 elpreima 6013 . . . . . 6  |-  ( F  Fn  ( Base `  S
)  ->  ( (
a ( .s `  S ) b )  e.  ( `' F " U )  <->  ( (
a ( .s `  S ) b )  e.  ( Base `  S
)  /\  ( F `  ( a ( .s
`  S ) b ) )  e.  U
) ) )
5216, 50, 513syl 18 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (
( a ( .s
`  S ) b )  e.  ( `' F " U )  <-> 
( ( a ( .s `  S ) b )  e.  (
Base `  S )  /\  ( F `  (
a ( .s `  S ) b ) )  e.  U ) ) )
5352adantr 466 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( ( a ( .s `  S
) b )  e.  ( `' F " U )  <->  ( (
a ( .s `  S ) b )  e.  ( Base `  S
)  /\  ( F `  ( a ( .s
`  S ) b ) )  e.  U
) ) )
5426, 49, 53mpbir2and 930 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( a ( .s `  S ) b )  e.  ( `' F " U ) )
5554ralrimivva 2846 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  A. a  e.  ( Base `  (Scalar `  S ) ) A. b  e.  ( `' F " U ) ( a ( .s `  S ) b )  e.  ( `' F " U ) )
569adantr 466 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  S  e.  LMod )
57 lmhmima.x . . . 4  |-  X  =  ( LSubSp `  S )
5822, 24, 13, 23, 57islss4 18172 . . 3  |-  ( S  e.  LMod  ->  ( ( `' F " U )  e.  X  <->  ( ( `' F " U )  e.  (SubGrp `  S
)  /\  A. a  e.  ( Base `  (Scalar `  S ) ) A. b  e.  ( `' F " U ) ( a ( .s `  S ) b )  e.  ( `' F " U ) ) ) )
5956, 58syl 17 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (
( `' F " U )  e.  X  <->  ( ( `' F " U )  e.  (SubGrp `  S )  /\  A. a  e.  ( Base `  (Scalar `  S )
) A. b  e.  ( `' F " U ) ( a ( .s `  S
) b )  e.  ( `' F " U ) ) ) )
608, 55, 59mpbir2and 930 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   `'ccnv 4848   dom cdm 4849   "cima 4852   Fun wfun 5591    Fn wfn 5592   -->wf 5593   ` cfv 5597  (class class class)co 6301   Basecbs 15108  Scalarcsca 15180   .scvsca 15181  SubGrpcsubg 16798    GrpHom cghm 16867   LModclmod 18078   LSubSpclss 18142   LMHom clmhm 18229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-0g 15327  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-grp 16660  df-minusg 16661  df-sbg 16662  df-subg 16801  df-ghm 16868  df-mgp 17711  df-ur 17723  df-ring 17769  df-lmod 18080  df-lss 18143  df-lmhm 18232
This theorem is referenced by:  lmhmlsp  18259  lmhmkerlss  18261  lnmepi  35862
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