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Theorem lmhmima 16115
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 16099 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
21adantr 452 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F  e.  ( S  GrpHom  T ) )
3 lmhmlmod1 16101 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
43adantr 452 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  S  e.  LMod )
5 simpr 448 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  e.  X )
6 lmhmima.x . . . . 5  |-  X  =  ( LSubSp `  S )
76lsssubg 16025 . . . 4  |-  ( ( S  e.  LMod  /\  U  e.  X )  ->  U  e.  (SubGrp `  S )
)
84, 5, 7syl2anc 643 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  e.  (SubGrp `  S )
)
9 ghmima 15018 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (SubGrp `  S )
)  ->  ( F " U )  e.  (SubGrp `  T ) )
102, 8, 9syl2anc 643 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( F " U )  e.  (SubGrp `  T )
)
11 eqid 2435 . . . . . . . . . 10  |-  ( Base `  S )  =  (
Base `  S )
12 eqid 2435 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
1311, 12lmhmf 16102 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
1413adantr 452 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F : ( Base `  S
) --> ( Base `  T
) )
15 ffn 5583 . . . . . . . 8  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1614, 15syl 16 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F  Fn  ( Base `  S
) )
1711, 6lssss 16005 . . . . . . . 8  |-  ( U  e.  X  ->  U  C_  ( Base `  S
) )
185, 17syl 16 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  C_  ( Base `  S
) )
19 fvelimab 5774 . . . . . . 7  |-  ( ( F  Fn  ( Base `  S )  /\  U  C_  ( Base `  S
) )  ->  (
b  e.  ( F
" U )  <->  E. c  e.  U  ( F `  c )  =  b ) )
2016, 18, 19syl2anc 643 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (
b  e.  ( F
" U )  <->  E. c  e.  U  ( F `  c )  =  b ) )
2120adantr 452 . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  F  e.  ( S LMHom  T ) )
23 eqid 2435 . . . . . . . . . . . . . . . 16  |-  (Scalar `  S )  =  (Scalar `  S )
24 eqid 2435 . . . . . . . . . . . . . . . 16  |-  (Scalar `  T )  =  (Scalar `  T )
2523, 24lmhmsca 16098 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
2625adantr 452 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (Scalar `  T )  =  (Scalar `  S ) )
2726fveq2d 5724 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
2827eleq2d 2502 . . . . . . . . . . . 12  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (
a  e.  ( Base `  (Scalar `  T )
)  <->  a  e.  (
Base `  (Scalar `  S
) ) ) )
2928biimpa 471 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  a  e.  ( Base `  (Scalar `  T
) ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
3029adantrr 698 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
3118sselda 3340 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  c  e.  U
)  ->  c  e.  ( Base `  S )
)
3231adantrl 697 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
c  e.  ( Base `  S ) )
33 eqid 2435 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
34 eqid 2435 . . . . . . . . . . 11  |-  ( .s
`  S )  =  ( .s `  S
)
35 eqid 2435 . . . . . . . . . . 11  |-  ( .s
`  T )  =  ( .s `  T
)
3623, 33, 11, 34, 35lmhmlin 16103 . . . . . . . . . 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 1184 . . . . . . . . 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 19 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  F  Fn  ( Base `  S ) )
39 simplr 732 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  U  e.  X )
4039, 17syl 16 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  U  C_  ( Base `  S
) )
414adantr 452 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  S  e.  LMod )
42 simprr 734 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
c  e.  U )
4323, 34, 33, 6lssvscl 16023 . . . . . . . . . . 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 1185 . . . . . . . . . 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 5968 . . . . . . . . . 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 1184 . . . . . . . . 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 2510 . . . . . . . 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 630 . . . . . . 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 6081 . . . . . . . 8  |-  ( ( F `  c )  =  b  ->  (
a ( .s `  T ) ( F `
 c ) )  =  ( a ( .s `  T ) b ) )
5049eleq1d 2501 . . . . . . 7  |-  ( ( F `  c )  =  b  ->  (
( a ( .s
`  T ) ( F `  c ) )  e.  ( F
" U )  <->  ( a
( .s `  T
) b )  e.  ( F " U
) ) )
5148, 50syl5ibcom 212 . . . . . 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 2822 . . . . 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 207 . . . 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 603 . . 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 2790 . 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 16100 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
5756adantr 452 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  T  e.  LMod )
58 eqid 2435 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
59 lmhmima.y . . . 4  |-  Y  =  ( LSubSp `  T )
6024, 58, 12, 35, 59islss4 16030 . . 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 16 . 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 889 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( F " U )  e.  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525  SubGrpcsubg 14930    GrpHom cghm 14995   LModclmod 15942   LSubSpclss 16000   LMHom clmhm 16087
This theorem is referenced by:  lmhmlsp  16117  lmhmrnlss  16118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lss 16001  df-lmhm 16090
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