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Theorem lmhmima 15820
Description: The 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
lmhmima  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( F " U )  e.  Y )

Proof of Theorem lmhmima
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 15804 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
21adantr 451 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F  e.  ( S  GrpHom  T ) )
3 lmhmlmod1 15806 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
43adantr 451 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  S  e.  LMod )
5 simpr 447 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  e.  X )
6 lmhmima.x . . . . 5  |-  X  =  ( LSubSp `  S )
76lsssubg 15730 . . . 4  |-  ( ( S  e.  LMod  /\  U  e.  X )  ->  U  e.  (SubGrp `  S )
)
84, 5, 7syl2anc 642 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  e.  (SubGrp `  S )
)
9 ghmima 14719 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (SubGrp `  S )
)  ->  ( F " U )  e.  (SubGrp `  T ) )
102, 8, 9syl2anc 642 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( F " U )  e.  (SubGrp `  T )
)
11 eqid 2296 . . . . . . . . . 10  |-  ( Base `  S )  =  (
Base `  S )
12 eqid 2296 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
1311, 12lmhmf 15807 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
1413adantr 451 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F : ( Base `  S
) --> ( Base `  T
) )
15 ffn 5405 . . . . . . . 8  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1614, 15syl 15 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F  Fn  ( Base `  S
) )
1711, 6lssss 15710 . . . . . . . 8  |-  ( U  e.  X  ->  U  C_  ( Base `  S
) )
185, 17syl 15 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  C_  ( Base `  S
) )
19 fvelimab 5594 . . . . . . 7  |-  ( ( F  Fn  ( Base `  S )  /\  U  C_  ( Base `  S
) )  ->  (
b  e.  ( F
" U )  <->  E. c  e.  U  ( F `  c )  =  b ) )
2016, 18, 19syl2anc 642 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (
b  e.  ( F
" U )  <->  E. c  e.  U  ( F `  c )  =  b ) )
2120adantr 451 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  a  e.  ( Base `  (Scalar `  T
) ) )  -> 
( b  e.  ( F " U )  <->  E. c  e.  U  ( F `  c )  =  b ) )
22 simpll 730 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  F  e.  ( S LMHom  T ) )
23 eqid 2296 . . . . . . . . . . . . . . . 16  |-  (Scalar `  S )  =  (Scalar `  S )
24 eqid 2296 . . . . . . . . . . . . . . . 16  |-  (Scalar `  T )  =  (Scalar `  T )
2523, 24lmhmsca 15803 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
2625adantr 451 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (Scalar `  T )  =  (Scalar `  S ) )
2726fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
2827eleq2d 2363 . . . . . . . . . . . 12  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (
a  e.  ( Base `  (Scalar `  T )
)  <->  a  e.  (
Base `  (Scalar `  S
) ) ) )
2928biimpa 470 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  a  e.  ( Base `  (Scalar `  T
) ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
3029adantrr 697 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
3118sselda 3193 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  c  e.  U
)  ->  c  e.  ( Base `  S )
)
3231adantrl 696 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
c  e.  ( Base `  S ) )
33 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
34 eqid 2296 . . . . . . . . . . 11  |-  ( .s
`  S )  =  ( .s `  S
)
35 eqid 2296 . . . . . . . . . . 11  |-  ( .s
`  T )  =  ( .s `  T
)
3623, 33, 11, 34, 35lmhmlin 15808 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  c  e.  ( Base `  S ) )  -> 
( F `  (
a ( .s `  S ) c ) )  =  ( a ( .s `  T
) ( F `  c ) ) )
3722, 30, 32, 36syl3anc 1182 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
( F `  (
a ( .s `  S ) c ) )  =  ( a ( .s `  T
) ( F `  c ) ) )
3822, 13, 153syl 18 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  F  Fn  ( Base `  S ) )
39 simplr 731 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  U  e.  X )
4039, 17syl 15 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  U  C_  ( Base `  S
) )
414adantr 451 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  S  e.  LMod )
42 simprr 733 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
c  e.  U )
4323, 34, 33, 6lssvscl 15728 . . . . . . . . . . 11  |-  ( ( ( S  e.  LMod  /\  U  e.  X )  /\  ( a  e.  ( Base `  (Scalar `  S ) )  /\  c  e.  U )
)  ->  ( a
( .s `  S
) c )  e.  U )
4441, 39, 30, 42, 43syl22anc 1183 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
( a ( .s
`  S ) c )  e.  U )
45 fnfvima 5772 . . . . . . . . . 10  |-  ( ( F  Fn  ( Base `  S )  /\  U  C_  ( Base `  S
)  /\  ( a
( .s `  S
) c )  e.  U )  ->  ( F `  ( a
( .s `  S
) c ) )  e.  ( F " U ) )
4638, 40, 44, 45syl3anc 1182 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
( F `  (
a ( .s `  S ) c ) )  e.  ( F
" U ) )
4737, 46eqeltrrd 2371 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
( a ( .s
`  T ) ( F `  c ) )  e.  ( F
" U ) )
4847anassrs 629 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X
)  /\  a  e.  ( Base `  (Scalar `  T
) ) )  /\  c  e.  U )  ->  ( a ( .s
`  T ) ( F `  c ) )  e.  ( F
" U ) )
49 oveq2 5882 . . . . . . . 8  |-  ( ( F `  c )  =  b  ->  (
a ( .s `  T ) ( F `
 c ) )  =  ( a ( .s `  T ) b ) )
5049eleq1d 2362 . . . . . . 7  |-  ( ( F `  c )  =  b  ->  (
( a ( .s
`  T ) ( F `  c ) )  e.  ( F
" U )  <->  ( a
( .s `  T
) b )  e.  ( F " U
) ) )
5148, 50syl5ibcom 211 . . . . . 6  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X
)  /\  a  e.  ( Base `  (Scalar `  T
) ) )  /\  c  e.  U )  ->  ( ( F `  c )  =  b  ->  ( a ( .s `  T ) b )  e.  ( F " U ) ) )
5251rexlimdva 2680 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  a  e.  ( Base `  (Scalar `  T
) ) )  -> 
( E. c  e.  U  ( F `  c )  =  b  ->  ( a ( .s `  T ) b )  e.  ( F " U ) ) )
5321, 52sylbid 206 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  a  e.  ( Base `  (Scalar `  T
) ) )  -> 
( b  e.  ( F " U )  ->  ( a ( .s `  T ) b )  e.  ( F " U ) ) )
5453impr 602 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  b  e.  ( F " U
) ) )  -> 
( a ( .s
`  T ) b )  e.  ( F
" U ) )
5554ralrimivva 2648 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  A. a  e.  ( Base `  (Scalar `  T ) ) A. b  e.  ( F " U ) ( a ( .s `  T
) b )  e.  ( F " U
) )
56 lmhmlmod2 15805 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
5756adantr 451 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  T  e.  LMod )
58 eqid 2296 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
59 lmhmima.y . . . 4  |-  Y  =  ( LSubSp `  T )
6024, 58, 12, 35, 59islss4 15735 . . 3  |-  ( T  e.  LMod  ->  ( ( F " U )  e.  Y  <->  ( ( F " U )  e.  (SubGrp `  T )  /\  A. a  e.  (
Base `  (Scalar `  T
) ) A. b  e.  ( F " U
) ( a ( .s `  T ) b )  e.  ( F " U ) ) ) )
6157, 60syl 15 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (
( F " U
)  e.  Y  <->  ( ( F " U )  e.  (SubGrp `  T )  /\  A. a  e.  (
Base `  (Scalar `  T
) ) A. b  e.  ( F " U
) ( a ( .s `  T ) b )  e.  ( F " U ) ) ) )
6210, 55, 61mpbir2and 888 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( F " U )  e.  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228  SubGrpcsubg 14631    GrpHom cghm 14696   LModclmod 15643   LSubSpclss 15705   LMHom clmhm 15792
This theorem is referenced by:  lmhmlsp  15822  lmhmrnlss  15823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lmhm 15795
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