MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmpreima Structured version   Unicode version

Theorem lmhmpreima 16126
Description: The inverse 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
lmhmpreima  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  X )

Proof of Theorem lmhmpreima
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 16109 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
21adantr 453 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  F  e.  ( S  GrpHom  T ) )
3 lmhmlmod2 16110 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
4 lmhmima.y . . . . 5  |-  Y  =  ( LSubSp `  T )
54lsssubg 16035 . . . 4  |-  ( ( T  e.  LMod  /\  U  e.  Y )  ->  U  e.  (SubGrp `  T )
)
63, 5sylan 459 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  U  e.  (SubGrp `  T )
)
7 ghmpreima 15029 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  U  e.  (SubGrp `  T )
)  ->  ( `' F " U )  e.  (SubGrp `  S )
)
82, 6, 7syl2anc 644 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  (SubGrp `  S
) )
9 lmhmlmod1 16111 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
109ad2antrr 708 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  S  e.  LMod )
11 simprl 734 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  a  e.  (
Base `  (Scalar `  S
) ) )
12 cnvimass 5226 . . . . . . . 8  |-  ( `' F " U ) 
C_  dom  F
13 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 16112 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
1615adantr 453 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  F : ( Base `  S
) --> ( Base `  T
) )
17 fdm 5597 . . . . . . . . 9  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
1816, 17syl 16 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  dom  F  =  ( Base `  S
) )
1912, 18syl5sseq 3398 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U ) 
C_  ( Base `  S
) )
2019sselda 3350 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  b  e.  ( `' F " U ) )  ->  b  e.  ( Base `  S )
)
2120adantrl 698 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  b  e.  (
Base `  S )
)
22 eqid 2438 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
23 eqid 2438 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
24 eqid 2438 . . . . . 6  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
2513, 22, 23, 24lmodvscl 15969 . . . . 5  |-  ( ( S  e.  LMod  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  ( Base `  S ) )  -> 
( a ( .s
`  S ) b )  e.  ( Base `  S ) )
2610, 11, 21, 25syl3anc 1185 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( a ( .s `  S ) b )  e.  (
Base `  S )
)
27 simpll 732 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  F  e.  ( S LMHom  T ) )
28 eqid 2438 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
2922, 24, 13, 23, 28lmhmlin 16113 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  ( Base `  S ) )  -> 
( F `  (
a ( .s `  S ) b ) )  =  ( a ( .s `  T
) ( F `  b ) ) )
3027, 11, 21, 29syl3anc 1185 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( F `  ( a ( .s
`  S ) b ) )  =  ( a ( .s `  T ) ( F `
 b ) ) )
313ad2antrr 708 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  T  e.  LMod )
32 simplr 733 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  U  e.  Y
)
33 eqid 2438 . . . . . . . . . . . 12  |-  (Scalar `  T )  =  (Scalar `  T )
3422, 33lmhmsca 16108 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
3534adantr 453 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (Scalar `  T )  =  (Scalar `  S ) )
3635fveq2d 5734 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( Base `  (Scalar `  T
) )  =  (
Base `  (Scalar `  S
) ) )
3736eleq2d 2505 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (
a  e.  ( Base `  (Scalar `  T )
)  <->  a  e.  (
Base `  (Scalar `  S
) ) ) )
3837biimpar 473 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  a  e.  ( Base `  (Scalar `  S
) ) )  -> 
a  e.  ( Base `  (Scalar `  T )
) )
3938adantrr 699 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  a  e.  (
Base `  (Scalar `  T
) ) )
40 ffun 5595 . . . . . . . . 9  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Fun  F )
4116, 40syl 16 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  Fun  F )
4241adantr 453 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  Fun  F )
43 simprr 735 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  b  e.  ( `' F " U ) )
44 fvimacnvi 5846 . . . . . . 7  |-  ( ( Fun  F  /\  b  e.  ( `' F " U ) )  -> 
( F `  b
)  e.  U )
4542, 43, 44syl2anc 644 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( F `  b )  e.  U
)
46 eqid 2438 . . . . . . 7  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
4733, 28, 46, 4lssvscl 16033 . . . . . 6  |-  ( ( ( T  e.  LMod  /\  U  e.  Y )  /\  ( a  e.  ( Base `  (Scalar `  T ) )  /\  ( F `  b )  e.  U ) )  ->  ( a ( .s `  T ) ( F `  b
) )  e.  U
)
4831, 32, 39, 45, 47syl22anc 1186 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( a ( .s `  T ) ( F `  b
) )  e.  U
)
4930, 48eqeltrd 2512 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( F `  ( a ( .s
`  S ) b ) )  e.  U
)
50 ffn 5593 . . . . . 6  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
51 elpreima 5852 . . . . . 6  |-  ( F  Fn  ( Base `  S
)  ->  ( (
a ( .s `  S ) b )  e.  ( `' F " U )  <->  ( (
a ( .s `  S ) b )  e.  ( Base `  S
)  /\  ( F `  ( a ( .s
`  S ) b ) )  e.  U
) ) )
5216, 50, 513syl 19 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (
( a ( .s
`  S ) b )  e.  ( `' F " U )  <-> 
( ( a ( .s `  S ) b )  e.  (
Base `  S )  /\  ( F `  (
a ( .s `  S ) b ) )  e.  U ) ) )
5352adantr 453 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( ( a ( .s `  S
) b )  e.  ( `' F " U )  <->  ( (
a ( .s `  S ) b )  e.  ( Base `  S
)  /\  ( F `  ( a ( .s
`  S ) b ) )  e.  U
) ) )
5426, 49, 53mpbir2and 890 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( `' F " U ) ) )  ->  ( a ( .s `  S ) b )  e.  ( `' F " U ) )
5554ralrimivva 2800 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  A. a  e.  ( Base `  (Scalar `  S ) ) A. b  e.  ( `' F " U ) ( a ( .s `  S ) b )  e.  ( `' F " U ) )
569adantr 453 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  S  e.  LMod )
57 lmhmima.x . . . 4  |-  X  =  ( LSubSp `  S )
5822, 24, 13, 23, 57islss4 16040 . . 3  |-  ( S  e.  LMod  ->  ( ( `' F " U )  e.  X  <->  ( ( `' F " U )  e.  (SubGrp `  S
)  /\  A. a  e.  ( Base `  (Scalar `  S ) ) A. b  e.  ( `' F " U ) ( a ( .s `  S ) b )  e.  ( `' F " U ) ) ) )
5956, 58syl 16 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  (
( `' F " U )  e.  X  <->  ( ( `' F " U )  e.  (SubGrp `  S )  /\  A. a  e.  ( Base `  (Scalar `  S )
) A. b  e.  ( `' F " U ) ( a ( .s `  S
) b )  e.  ( `' F " U ) ) ) )
608, 55, 59mpbir2and 890 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   `'ccnv 4879   dom cdm 4880   "cima 4883   Fun wfun 5450    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .scvsca 13535  SubGrpcsubg 14940    GrpHom cghm 15005   LModclmod 15952   LSubSpclss 16010   LMHom clmhm 16097
This theorem is referenced by:  lmhmlsp  16127  lmhmkerlss  16129  lnmepi  27162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-ghm 15006  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lss 16011  df-lmhm 16100
  Copyright terms: Public domain W3C validator