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Theorem lmhmima 17126
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 17110 . . . 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 17112 . . . . 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 17036 . . . 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 15765 . . 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 2441 . . . . . . . . . 10  |-  ( Base `  S )  =  (
Base `  S )
12 eqid 2441 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
1311, 12lmhmf 17113 . . . . . . . . 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 5557 . . . . . . . 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 17016 . . . . . . . 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 5745 . . . . . . 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 2441 . . . . . . . . . . . . . . . 16  |-  (Scalar `  S )  =  (Scalar `  S )
24 eqid 2441 . . . . . . . . . . . . . . . 16  |-  (Scalar `  T )  =  (Scalar `  T )
2523, 24lmhmsca 17109 . . . . . . . . . . . . . . 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 5693 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
2827eleq2d 2508 . . . . . . . . . . . 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 3354 . . . . . . . . . . 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 2441 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
34 eqid 2441 . . . . . . . . . . 11  |-  ( .s
`  S )  =  ( .s `  S
)
35 eqid 2441 . . . . . . . . . . 11  |-  ( .s
`  T )  =  ( .s `  T
)
3623, 33, 11, 34, 35lmhmlin 17114 . . . . . . . . . 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 1218 . . . . . . . . 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 17034 . . . . . . . . . . 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 1219 . . . . . . . . . 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 5953 . . . . . . . . . 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 1218 . . . . . . . . 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 2516 . . . . . . . 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 6097 . . . . . . . 8  |-  ( ( F `  c )  =  b  ->  (
a ( .s `  T ) ( F `
 c ) )  =  ( a ( .s `  T ) b ) )
5049eleq1d 2507 . . . . . . 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 2839 . . . . 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 2806 . 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 17111 . . . 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 2441 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
59 lmhmima.y . . . 4  |-  Y  =  ( LSubSp `  T )
6024, 58, 12, 35, 59islss4 17041 . . 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 913 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 1369    e. wcel 1756   A.wral 2713   E.wrex 2714    C_ wss 3326   "cima 4841    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089   Basecbs 14172  Scalarcsca 14239   .scvsca 14240  SubGrpcsubg 15673    GrpHom cghm 15742   LModclmod 16946   LSubSpclss 17011   LMHom clmhm 17098
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545  df-subg 15676  df-ghm 15743  df-mgp 16590  df-ur 16602  df-rng 16645  df-lmod 16948  df-lss 17012  df-lmhm 17101
This theorem is referenced by:  lmhmlsp  17128  lmhmrnlss  17129
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