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Theorem lmhmpreima 17565
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 17548 . . . 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 17549 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
4 lmhmima.y . . . . 5  |-  Y  =  ( LSubSp `  T )
54lsssubg 17474 . . . 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 16160 . . 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 17550 . . . . . 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 5363 . . . . . . . 8  |-  ( `' F " U ) 
C_  dom  F
13 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 17551 . . . . . . . . . 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 5741 . . . . . . . . 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 3557 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U ) 
C_  ( Base `  S
) )
2019sselda 3509 . . . . . 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 2467 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
23 eqid 2467 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
24 eqid 2467 . . . . . 6  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
2513, 22, 23, 24lmodvscl 17400 . . . . 5  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  ( Base `  S ) )  -> 
( a ( .s
`  S ) b )  e.  ( Base `  S ) )
2610, 11, 21, 25syl3anc 1228 . . . 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 2467 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
2922, 24, 13, 23, 28lmhmlin 17552 . . . . . 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 1228 . . . . 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 2467 . . . . . . . . . . . 12  |-  (Scalar `  T )  =  (Scalar `  T )
3422, 33lmhmsca 17547 . . . . . . . . . . 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 5876 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
3736eleq2d 2537 . . . . . . . 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 5739 . . . . . . . . 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 6002 . . . . . . 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 2467 . . . . . . 7  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
4733, 28, 46, 4lssvscl 17472 . . . . . 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 1229 . . . . 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 2555 . . . 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 5737 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
51 elpreima 6008 . . . . . 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 920 . . 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 2888 . 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 17479 . . 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 920 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 1379    e. wcel 1767   A.wral 2817   `'ccnv 5004   dom cdm 5005   "cima 5008   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507  Scalarcsca 14575   .scvsca 14576  SubGrpcsubg 16067    GrpHom cghm 16136   LModclmod 17383   LSubSpclss 17449   LMHom clmhm 17536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-lmod 17385  df-lss 17450  df-lmhm 17539
This theorem is referenced by:  lmhmlsp  17566  lmhmkerlss  17568  lnmepi  30959
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