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Theorem lmhmeql 15828
Description: The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
lmhmeql.u  |-  U  =  ( LSubSp `  S )
Assertion
Ref Expression
lmhmeql  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  dom  ( F  i^i  G )  e.  U )

Proof of Theorem lmhmeql
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 15804 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2 lmghm 15804 . . 3  |-  ( G  e.  ( S LMHom  T
)  ->  G  e.  ( S  GrpHom  T ) )
3 ghmeql 14721 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
41, 2, 3syl2an 463 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
5 lmhmlmod1 15806 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
65adantr 451 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
76ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  S  e.  LMod )
8 simplr 731 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  x  e.  ( Base `  (Scalar `  S
) ) )
9 simprl 732 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  y  e.  ( Base `  S )
)
10 eqid 2296 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
11 eqid 2296 . . . . . . . . 9  |-  (Scalar `  S )  =  (Scalar `  S )
12 eqid 2296 . . . . . . . . 9  |-  ( .s
`  S )  =  ( .s `  S
)
13 eqid 2296 . . . . . . . . 9  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
1410, 11, 12, 13lmodvscl 15660 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) )  -> 
( x ( .s
`  S ) y )  e.  ( Base `  S ) )
157, 8, 9, 14syl3anc 1182 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  ( x
( .s `  S
) y )  e.  ( Base `  S
) )
16 oveq2 5882 . . . . . . . . 9  |-  ( ( F `  y )  =  ( G `  y )  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  T ) ( G `  y
) ) )
1716ad2antll 709 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  ( x
( .s `  T
) ( F `  y ) )  =  ( x ( .s
`  T ) ( G `  y ) ) )
18 simplll 734 . . . . . . . . 9  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  F  e.  ( S LMHom  T ) )
19 eqid 2296 . . . . . . . . . 10  |-  ( .s
`  T )  =  ( .s `  T
)
2011, 13, 10, 12, 19lmhmlin 15808 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) )  -> 
( F `  (
x ( .s `  S ) y ) )  =  ( x ( .s `  T
) ( F `  y ) ) )
2118, 8, 9, 20syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  ( F `  ( x ( .s
`  S ) y ) )  =  ( x ( .s `  T ) ( F `
 y ) ) )
22 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  G  e.  ( S LMHom  T ) )
2311, 13, 10, 12, 19lmhmlin 15808 . . . . . . . . 9  |-  ( ( G  e.  ( S LMHom 
T )  /\  x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) )  -> 
( G `  (
x ( .s `  S ) y ) )  =  ( x ( .s `  T
) ( G `  y ) ) )
2422, 8, 9, 23syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  ( G `  ( x ( .s
`  S ) y ) )  =  ( x ( .s `  T ) ( G `
 y ) ) )
2517, 21, 243eqtr4d 2338 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  ( F `  ( x ( .s
`  S ) y ) )  =  ( G `  ( x ( .s `  S
) y ) ) )
26 fveq2 5541 . . . . . . . . 9  |-  ( z  =  ( x ( .s `  S ) y )  ->  ( F `  z )  =  ( F `  ( x ( .s
`  S ) y ) ) )
27 fveq2 5541 . . . . . . . . 9  |-  ( z  =  ( x ( .s `  S ) y )  ->  ( G `  z )  =  ( G `  ( x ( .s
`  S ) y ) ) )
2826, 27eqeq12d 2310 . . . . . . . 8  |-  ( z  =  ( x ( .s `  S ) y )  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  ( x ( .s
`  S ) y ) )  =  ( G `  ( x ( .s `  S
) y ) ) ) )
2928elrab 2936 . . . . . . 7  |-  ( ( x ( .s `  S ) y )  e.  { z  e.  ( Base `  S
)  |  ( F `
 z )  =  ( G `  z
) }  <->  ( (
x ( .s `  S ) y )  e.  ( Base `  S
)  /\  ( F `  ( x ( .s
`  S ) y ) )  =  ( G `  ( x ( .s `  S
) y ) ) ) )
3015, 25, 29sylanbrc 645 . . . . . 6  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  ( y  e.  (
Base `  S )  /\  ( F `  y
)  =  ( G `
 y ) ) )  ->  ( x
( .s `  S
) y )  e. 
{ z  e.  (
Base `  S )  |  ( F `  z )  =  ( G `  z ) } )
3130expr 598 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  (
Base `  (Scalar `  S
) ) )  /\  y  e.  ( Base `  S ) )  -> 
( ( F `  y )  =  ( G `  y )  ->  ( x ( .s `  S ) y )  e.  {
z  e.  ( Base `  S )  |  ( F `  z )  =  ( G `  z ) } ) )
3231ralrimiva 2639 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  ( Base `  (Scalar `  S ) ) )  ->  A. y  e.  (
Base `  S )
( ( F `  y )  =  ( G `  y )  ->  ( x ( .s `  S ) y )  e.  {
z  e.  ( Base `  S )  |  ( F `  z )  =  ( G `  z ) } ) )
33 eqid 2296 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
3410, 33lmhmf 15807 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
35 ffn 5405 . . . . . . . 8  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
3634, 35syl 15 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  F  Fn  ( Base `  S )
)
3710, 33lmhmf 15807 . . . . . . . 8  |-  ( G  e.  ( S LMHom  T
)  ->  G :
( Base `  S ) --> ( Base `  T )
)
38 ffn 5405 . . . . . . . 8  |-  ( G : ( Base `  S
) --> ( Base `  T
)  ->  G  Fn  ( Base `  S )
)
3937, 38syl 15 . . . . . . 7  |-  ( G  e.  ( S LMHom  T
)  ->  G  Fn  ( Base `  S )
)
40 fndmin 5648 . . . . . . 7  |-  ( ( F  Fn  ( Base `  S )  /\  G  Fn  ( Base `  S
) )  ->  dom  ( F  i^i  G )  =  { z  e.  ( Base `  S
)  |  ( F `
 z )  =  ( G `  z
) } )
4136, 39, 40syl2an 463 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  dom  ( F  i^i  G )  =  { z  e.  (
Base `  S )  |  ( F `  z )  =  ( G `  z ) } )
4241adantr 451 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  ( Base `  (Scalar `  S ) ) )  ->  dom  ( F  i^i  G )  =  {
z  e.  ( Base `  S )  |  ( F `  z )  =  ( G `  z ) } )
43 eleq2 2357 . . . . . . 7  |-  ( dom  ( F  i^i  G
)  =  { z  e.  ( Base `  S
)  |  ( F `
 z )  =  ( G `  z
) }  ->  (
( x ( .s
`  S ) y )  e.  dom  ( F  i^i  G )  <->  ( x
( .s `  S
) y )  e. 
{ z  e.  (
Base `  S )  |  ( F `  z )  =  ( G `  z ) } ) )
4443raleqbi1dv 2757 . . . . . 6  |-  ( dom  ( F  i^i  G
)  =  { z  e.  ( Base `  S
)  |  ( F `
 z )  =  ( G `  z
) }  ->  ( A. y  e.  dom  ( F  i^i  G ) ( x ( .s
`  S ) y )  e.  dom  ( F  i^i  G )  <->  A. y  e.  { z  e.  (
Base `  S )  |  ( F `  z )  =  ( G `  z ) }  ( x ( .s `  S ) y )  e.  {
z  e.  ( Base `  S )  |  ( F `  z )  =  ( G `  z ) } ) )
45 fveq2 5541 . . . . . . . 8  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
46 fveq2 5541 . . . . . . . 8  |-  ( z  =  y  ->  ( G `  z )  =  ( G `  y ) )
4745, 46eqeq12d 2310 . . . . . . 7  |-  ( z  =  y  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  y )  =  ( G `  y ) ) )
4847ralrab 2940 . . . . . 6  |-  ( A. y  e.  { z  e.  ( Base `  S
)  |  ( F `
 z )  =  ( G `  z
) }  ( x ( .s `  S
) y )  e. 
{ z  e.  (
Base `  S )  |  ( F `  z )  =  ( G `  z ) }  <->  A. y  e.  (
Base `  S )
( ( F `  y )  =  ( G `  y )  ->  ( x ( .s `  S ) y )  e.  {
z  e.  ( Base `  S )  |  ( F `  z )  =  ( G `  z ) } ) )
4944, 48syl6bb 252 . . . . 5  |-  ( dom  ( F  i^i  G
)  =  { z  e.  ( Base `  S
)  |  ( F `
 z )  =  ( G `  z
) }  ->  ( A. y  e.  dom  ( F  i^i  G ) ( x ( .s
`  S ) y )  e.  dom  ( F  i^i  G )  <->  A. y  e.  ( Base `  S
) ( ( F `
 y )  =  ( G `  y
)  ->  ( x
( .s `  S
) y )  e. 
{ z  e.  (
Base `  S )  |  ( F `  z )  =  ( G `  z ) } ) ) )
5042, 49syl 15 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  ( Base `  (Scalar `  S ) ) )  ->  ( A. y  e.  dom  ( F  i^i  G ) ( x ( .s `  S ) y )  e.  dom  ( F  i^i  G )  <->  A. y  e.  ( Base `  S ) ( ( F `  y
)  =  ( G `
 y )  -> 
( x ( .s
`  S ) y )  e.  { z  e.  ( Base `  S
)  |  ( F `
 z )  =  ( G `  z
) } ) ) )
5132, 50mpbird 223 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  /\  x  e.  ( Base `  (Scalar `  S ) ) )  ->  A. y  e.  dom  ( F  i^i  G ) ( x ( .s
`  S ) y )  e.  dom  ( F  i^i  G ) )
5251ralrimiva 2639 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  A. x  e.  ( Base `  (Scalar `  S ) ) A. y  e.  dom  ( F  i^i  G ) ( x ( .s `  S ) y )  e.  dom  ( F  i^i  G ) )
53 lmhmeql.u . . . 4  |-  U  =  ( LSubSp `  S )
5411, 13, 10, 12, 53islss4 15735 . . 3  |-  ( S  e.  LMod  ->  ( dom  ( F  i^i  G
)  e.  U  <->  ( dom  ( F  i^i  G )  e.  (SubGrp `  S
)  /\  A. x  e.  ( Base `  (Scalar `  S ) ) A. y  e.  dom  ( F  i^i  G ) ( x ( .s `  S ) y )  e.  dom  ( F  i^i  G ) ) ) )
556, 54syl 15 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  ( dom  ( F  i^i  G )  e.  U  <->  ( dom  ( F  i^i  G )  e.  (SubGrp `  S
)  /\  A. x  e.  ( Base `  (Scalar `  S ) ) A. y  e.  dom  ( F  i^i  G ) ( x ( .s `  S ) y )  e.  dom  ( F  i^i  G ) ) ) )
564, 52, 55mpbir2and 888 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  dom  ( F  i^i  G )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    i^i cin 3164   dom cdm 4705    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228  SubGrpcsubg 14631    GrpHom cghm 14696   LModclmod 15643   LSubSpclss 15705   LMHom clmhm 15792
This theorem is referenced by:  lspextmo  15829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lmhm 15795
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