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Theorem lmhmeql 15812
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 15788 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2 lmghm 15788 . . 3  |-  ( G  e.  ( S LMHom  T
)  ->  G  e.  ( S  GrpHom  T ) )
3 ghmeql 14705 . . 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 15790 . . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
11 eqid 2283 . . . . . . . . 9  |-  (Scalar `  S )  =  (Scalar `  S )
12 eqid 2283 . . . . . . . . 9  |-  ( .s
`  S )  =  ( .s `  S
)
13 eqid 2283 . . . . . . . . 9  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
1410, 11, 12, 13lmodvscl 15644 . . . . . . . 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 5866 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  ( .s
`  T )  =  ( .s `  T
)
2011, 13, 10, 12, 19lmhmlin 15792 . . . . . . . . 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 15792 . . . . . . . . 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 2325 . . . . . . 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 5525 . . . . . . . . 9  |-  ( z  =  ( x ( .s `  S ) y )  ->  ( F `  z )  =  ( F `  ( x ( .s
`  S ) y ) ) )
27 fveq2 5525 . . . . . . . . 9  |-  ( z  =  ( x ( .s `  S ) y )  ->  ( G `  z )  =  ( G `  ( x ( .s
`  S ) y ) ) )
2826, 27eqeq12d 2297 . . . . . . . 8  |-  ( z  =  ( x ( .s `  S ) y )  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  ( x ( .s
`  S ) y ) )  =  ( G `  ( x ( .s `  S
) y ) ) ) )
2928elrab 2923 . . . . . . 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 2626 . . . 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 2283 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
3410, 33lmhmf 15791 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
35 ffn 5389 . . . . . . . 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 15791 . . . . . . . 8  |-  ( G  e.  ( S LMHom  T
)  ->  G :
( Base `  S ) --> ( Base `  T )
)
38 ffn 5389 . . . . . . . 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 5632 . . . . . . 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 2344 . . . . . . 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 2744 . . . . . 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 5525 . . . . . . . 8  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
46 fveq2 5525 . . . . . . . 8  |-  ( z  =  y  ->  ( G `  z )  =  ( G `  y ) )
4745, 46eqeq12d 2297 . . . . . . 7  |-  ( z  =  y  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  y )  =  ( G `  y ) ) )
4847ralrab 2927 . . . . . 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 2626 . 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 15719 . . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547    i^i cin 3151   dom cdm 4689    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:  lspextmo  15813
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-map 6774  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-mhm 14415  df-submnd 14416  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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