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Theorem lmhmeql 16131
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 16107 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2 lmghm 16107 . . 3  |-  ( G  e.  ( S LMHom  T
)  ->  G  e.  ( S  GrpHom  T ) )
3 ghmeql 15028 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  G  e.  ( S  GrpHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
41, 2, 3syl2an 464 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubGrp `  S )
)
5 lmhmlmod1 16109 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
65adantr 452 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  G  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
76ad2antrr 707 . . . . . . . 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 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 ) ) )  ->  x  e.  ( Base `  (Scalar `  S
) ) )
9 simprl 733 . . . . . . . 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 2436 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
11 eqid 2436 . . . . . . . . 9  |-  (Scalar `  S )  =  (Scalar `  S )
12 eqid 2436 . . . . . . . . 9  |-  ( .s
`  S )  =  ( .s `  S
)
13 eqid 2436 . . . . . . . . 9  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
1410, 11, 12, 13lmodvscl 15967 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) )  -> 
( x ( .s
`  S ) y )  e.  ( Base `  S ) )
157, 8, 9, 14syl3anc 1184 . . . . . . 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 6089 . . . . . . . . 9  |-  ( ( F `  y )  =  ( G `  y )  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  T ) ( G `  y
) ) )
1716ad2antll 710 . . . . . . . 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 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 ) ) )  ->  F  e.  ( S LMHom  T ) )
19 eqid 2436 . . . . . . . . . 10  |-  ( .s
`  T )  =  ( .s `  T
)
2011, 13, 10, 12, 19lmhmlin 16111 . . . . . . . . 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 1184 . . . . . . . 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 736 . . . . . . . . 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 16111 . . . . . . . . 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 1184 . . . . . . . 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 2478 . . . . . . 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 5728 . . . . . . . . 9  |-  ( z  =  ( x ( .s `  S ) y )  ->  ( F `  z )  =  ( F `  ( x ( .s
`  S ) y ) ) )
27 fveq2 5728 . . . . . . . . 9  |-  ( z  =  ( x ( .s `  S ) y )  ->  ( G `  z )  =  ( G `  ( x ( .s
`  S ) y ) ) )
2826, 27eqeq12d 2450 . . . . . . . 8  |-  ( z  =  ( x ( .s `  S ) y )  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  ( x ( .s
`  S ) y ) )  =  ( G `  ( x ( .s `  S
) y ) ) ) )
2928elrab 3092 . . . . . . 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 646 . . . . . 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 599 . . . . 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 2789 . . . 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 2436 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
3410, 33lmhmf 16110 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
35 ffn 5591 . . . . . . . 8  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
3634, 35syl 16 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  F  Fn  ( Base `  S )
)
3710, 33lmhmf 16110 . . . . . . . 8  |-  ( G  e.  ( S LMHom  T
)  ->  G :
( Base `  S ) --> ( Base `  T )
)
38 ffn 5591 . . . . . . . 8  |-  ( G : ( Base `  S
) --> ( Base `  T
)  ->  G  Fn  ( Base `  S )
)
3937, 38syl 16 . . . . . . 7  |-  ( G  e.  ( S LMHom  T
)  ->  G  Fn  ( Base `  S )
)
40 fndmin 5837 . . . . . . 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 464 . . . . . 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 452 . . . . 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 2497 . . . . . . 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 2912 . . . . . 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 5728 . . . . . . . 8  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
46 fveq2 5728 . . . . . . . 8  |-  ( z  =  y  ->  ( G `  z )  =  ( G `  y ) )
4745, 46eqeq12d 2450 . . . . . . 7  |-  ( z  =  y  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  y )  =  ( G `  y ) ) )
4847ralrab 3096 . . . . . 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 253 . . . . 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 16 . . . 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 224 . . 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 2789 . 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 16038 . . 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 16 . 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 889 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709    i^i cin 3319   dom cdm 4878    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .scvsca 13533  SubGrpcsubg 14938    GrpHom cghm 15003   LModclmod 15950   LSubSpclss 16008   LMHom clmhm 16095
This theorem is referenced by:  lspextmo  16132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-ghm 15004  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-lmhm 16098
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