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Theorem lmhmima 17473
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 17457 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
21adantr 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F  e.  ( S  GrpHom  T ) )
3 lmhmlmod1 17459 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
43adantr 465 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  S  e.  LMod )
5 simpr 461 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  e.  X )
6 lmhmima.x . . . . 5  |-  X  =  ( LSubSp `  S )
76lsssubg 17383 . . . 4  |-  ( ( S  e.  LMod  /\  U  e.  X )  ->  U  e.  (SubGrp `  S )
)
84, 5, 7syl2anc 661 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  U  e.  (SubGrp `  S )
)
9 ghmima 16079 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (SubGrp `  S )
)  ->  ( F " U )  e.  (SubGrp `  T ) )
102, 8, 9syl2anc 661 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( F " U )  e.  (SubGrp `  T )
)
11 eqid 2467 . . . . . . . . . 10  |-  ( Base `  S )  =  (
Base `  S )
12 eqid 2467 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
1311, 12lmhmf 17460 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
1413adantr 465 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  F : ( Base `  S
) --> ( Base `  T
) )
15 ffn 5729 . . . . . . . 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 17363 . . . . . . . 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 5921 . . . . . . 7  |-  ( ( F  Fn  ( Base `  S )  /\  U  C_  ( Base `  S
) )  ->  (
b  e.  ( F
" U )  <->  E. c  e.  U  ( F `  c )  =  b ) )
2016, 18, 19syl2anc 661 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (
b  e.  ( F
" U )  <->  E. c  e.  U  ( F `  c )  =  b ) )
2120adantr 465 . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  F  e.  ( S LMHom  T ) )
23 eqid 2467 . . . . . . . . . . . . . . . 16  |-  (Scalar `  S )  =  (Scalar `  S )
24 eqid 2467 . . . . . . . . . . . . . . . 16  |-  (Scalar `  T )  =  (Scalar `  T )
2523, 24lmhmsca 17456 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
2625adantr 465 . . . . . . . . . . . . . 14  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (Scalar `  T )  =  (Scalar `  S ) )
2726fveq2d 5868 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
2827eleq2d 2537 . . . . . . . . . . . 12  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  (
a  e.  ( Base `  (Scalar `  T )
)  <->  a  e.  (
Base `  (Scalar `  S
) ) ) )
2928biimpa 484 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  a  e.  ( Base `  (Scalar `  T
) ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
3029adantrr 716 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
3118sselda 3504 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  c  e.  U
)  ->  c  e.  ( Base `  S )
)
3231adantrl 715 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
c  e.  ( Base `  S ) )
33 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
34 eqid 2467 . . . . . . . . . . 11  |-  ( .s
`  S )  =  ( .s `  S
)
35 eqid 2467 . . . . . . . . . . 11  |-  ( .s
`  T )  =  ( .s `  T
)
3623, 33, 11, 34, 35lmhmlin 17461 . . . . . . . . . 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 1228 . . . . . . . . 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 20 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  F  Fn  ( Base `  S ) )
39 simplr 754 . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  ->  S  e.  LMod )
42 simprr 756 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  /\  ( a  e.  (
Base `  (Scalar `  T
) )  /\  c  e.  U ) )  -> 
c  e.  U )
4323, 34, 33, 6lssvscl 17381 . . . . . . . . . . 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 1229 . . . . . . . . . 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 6136 . . . . . . . . . 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 1228 . . . . . . . . 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 2556 . . . . . . . 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 648 . . . . . . 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 6290 . . . . . . . 8  |-  ( ( F `  c )  =  b  ->  (
a ( .s `  T ) ( F `
 c ) )  =  ( a ( .s `  T ) b ) )
5049eleq1d 2536 . . . . . . 7  |-  ( ( F `  c )  =  b  ->  (
( a ( .s
`  T ) ( F `  c ) )  e.  ( F
" U )  <->  ( a
( .s `  T
) b )  e.  ( F " U
) ) )
5148, 50syl5ibcom 220 . . . . . 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 2955 . . . . 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 215 . . . 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 619 . . 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 2885 . 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 17458 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
5756adantr 465 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  T  e.  LMod )
58 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
59 lmhmima.y . . . 4  |-  Y  =  ( LSubSp `  T )
6024, 58, 12, 35, 59islss4 17388 . . 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 920 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( F " U )  e.  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14483  Scalarcsca 14551   .scvsca 14552  SubGrpcsubg 15987    GrpHom cghm 16056   LModclmod 17292   LSubSpclss 17358   LMHom clmhm 17445
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-sbg 15857  df-subg 15990  df-ghm 16057  df-mgp 16929  df-ur 16941  df-rng 16985  df-lmod 17294  df-lss 17359  df-lmhm 17448
This theorem is referenced by:  lmhmlsp  17475  lmhmrnlss  17476
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