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Theorem lmhmima 15804
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 15788 . . . 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 15790 . . . . 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 15714 . . . 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 14703 . . 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 2283 . . . . . . . . . 10  |-  ( Base `  S )  =  (
Base `  S )
12 eqid 2283 . . . . . . . . . 10  |-  ( Base `  T )  =  (
Base `  T )
1311, 12lmhmf 15791 . . . . . . . . 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 5389 . . . . . . . 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 15694 . . . . . . . 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 5578 . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  (Scalar `  S )  =  (Scalar `  S )
24 eqid 2283 . . . . . . . . . . . . . . . 16  |-  (Scalar `  T )  =  (Scalar `  T )
2523, 24lmhmsca 15787 . . . . . . . . . . . . . . 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 5529 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  X )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
2827eleq2d 2350 . . . . . . . . . . . 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 3180 . . . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
34 eqid 2283 . . . . . . . . . . 11  |-  ( .s
`  S )  =  ( .s `  S
)
35 eqid 2283 . . . . . . . . . . 11  |-  ( .s
`  T )  =  ( .s `  T
)
3623, 33, 11, 34, 35lmhmlin 15792 . . . . . . . . . 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 15712 . . . . . . . . . . 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 5756 . . . . . . . . . 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 2358 . . . . . . . 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 5866 . . . . . . . 8  |-  ( ( F `  c )  =  b  ->  (
a ( .s `  T ) ( F `
 c ) )  =  ( a ( .s `  T ) b ) )
5049eleq1d 2349 . . . . . . 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 2667 . . . . 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 2635 . 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 15789 . . . 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 2283 . . . 4  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
59 lmhmima.y . . . 4  |-  Y  =  ( LSubSp `  T )
6024, 58, 12, 35, 59islss4 15719 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212  SubGrpcsubg 14615    GrpHom cghm 14680   LModclmod 15627   LSubSpclss 15689   LMHom clmhm 15776
This theorem is referenced by:  lmhmlsp  15806  lmhmrnlss  15807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lmhm 15779
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