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Theorem lmhmpreima 17141
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 17124 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
21adantr 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  F  e.  ( S  GrpHom  T ) )
3 lmhmlmod2 17125 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
4 lmhmima.y . . . . 5  |-  Y  =  ( LSubSp `  T )
54lsssubg 17050 . . . 4  |-  ( ( T  e.  LMod  /\  U  e.  Y )  ->  U  e.  (SubGrp `  T )
)
63, 5sylan 471 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  U  e.  (SubGrp `  T )
)
7 ghmpreima 15780 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (SubGrp `  T )
)  ->  ( `' F " U )  e.  (SubGrp `  S )
)
82, 6, 7syl2anc 661 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  (SubGrp `  S
) )
9 lmhmlmod1 17126 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
109ad2antrr 725 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  S  e.  LMod )
11 simprl 755 . . . . 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 5201 . . . . . . . 8  |-  ( `' F " U ) 
C_  dom  F
13 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 17127 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
1615adantr 465 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  F : ( Base `  S
) --> ( Base `  T
) )
17 fdm 5575 . . . . . . . . 9  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
1816, 17syl 16 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  dom  F  =  ( Base `  S
) )
1912, 18syl5sseq 3416 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U ) 
C_  ( Base `  S
) )
2019sselda 3368 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  b  e.  ( `' F " U ) )  ->  b  e.  ( Base `  S )
)
2120adantrl 715 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  b  e.  (
Base `  S )
)
22 eqid 2443 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
23 eqid 2443 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
24 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
2513, 22, 23, 24lmodvscl 16977 . . . . 5  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  ( Base `  S ) )  -> 
( a ( .s
`  S ) b )  e.  ( Base `  S ) )
2610, 11, 21, 25syl3anc 1218 . . . 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 753 . . . . . 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 2443 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
2922, 24, 13, 23, 28lmhmlin 17128 . . . . . 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 1218 . . . . 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 725 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  T  e.  LMod )
32 simplr 754 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  U  e.  Y
)
33 eqid 2443 . . . . . . . . . . . 12  |-  (Scalar `  T )  =  (Scalar `  T )
3422, 33lmhmsca 17123 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
3534adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (Scalar `  T )  =  (Scalar `  S ) )
3635fveq2d 5707 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
3736eleq2d 2510 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (
a  e.  ( Base `  (Scalar `  T )
)  <->  a  e.  (
Base `  (Scalar `  S
) ) ) )
3837biimpar 485 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  a  e.  ( Base `  (Scalar `  S
) ) )  -> 
a  e.  ( Base `  (Scalar `  T )
) )
3938adantrr 716 . . . . . 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 5573 . . . . . . . . 9  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Fun  F )
4116, 40syl 16 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  Fun  F )
4241adantr 465 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  Fun  F )
43 simprr 756 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  b  e.  ( `' F " U ) )
44 fvimacnvi 5829 . . . . . . 7  |-  ( ( Fun  F  /\  b  e.  ( `' F " U ) )  -> 
( F `  b
)  e.  U )
4542, 43, 44syl2anc 661 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( F `  b )  e.  U
)
46 eqid 2443 . . . . . . 7  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
4733, 28, 46, 4lssvscl 17048 . . . . . 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 1219 . . . . 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 2517 . . . 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 5571 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
51 elpreima 5835 . . . . . 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 20 . . . . 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 465 . . . 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 913 . . 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 2820 . 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 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  S  e.  LMod )
57 lmhmima.x . . . 4  |-  X  =  ( LSubSp `  S )
5822, 24, 13, 23, 57islss4 17055 . . 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 16 . 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 913 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727   `'ccnv 4851   dom cdm 4852   "cima 4855   Fun wfun 5424    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   Basecbs 14186  Scalarcsca 14253   .scvsca 14254  SubGrpcsubg 15687    GrpHom cghm 15756   LModclmod 16960   LSubSpclss 17025   LMHom clmhm 17112
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 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-ghm 15757  df-mgp 16604  df-ur 16616  df-rng 16659  df-lmod 16962  df-lss 17026  df-lmhm 17115
This theorem is referenced by:  lmhmlsp  17142  lmhmkerlss  17144  lnmepi  29450
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