MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reslmhm2 Structured version   Unicode version

Theorem reslmhm2 17242
Description: Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Hypotheses
Ref Expression
reslmhm2.u  |-  U  =  ( Ts  X )
reslmhm2.l  |-  L  =  ( LSubSp `  T )
Assertion
Ref Expression
reslmhm2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )

Proof of Theorem reslmhm2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . 2  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2451 . 2  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2451 . 2  |-  ( .s
`  T )  =  ( .s `  T
)
4 eqid 2451 . 2  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2451 . 2  |-  (Scalar `  T )  =  (Scalar `  T )
6 eqid 2451 . 2  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 17222 . . 3  |-  ( F  e.  ( S LMHom  U
)  ->  S  e.  LMod )
873ad2ant1 1009 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  S  e.  LMod )
9 simp2 989 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  T  e.  LMod )
10 reslmhm2.u . . . . 5  |-  U  =  ( Ts  X )
1110, 5resssca 14420 . . . 4  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
12113ad2ant3 1011 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  U ) )
13 eqid 2451 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
144, 13lmhmsca 17219 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  (Scalar `  U
)  =  (Scalar `  S ) )
15143ad2ant1 1009 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  U )  =  (Scalar `  S ) )
1612, 15eqtrd 2492 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  S ) )
17 lmghm 17220 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  F  e.  ( S  GrpHom  U ) )
18173ad2ant1 1009 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S  GrpHom  U ) )
19 reslmhm2.l . . . . 5  |-  L  =  ( LSubSp `  T )
2019lsssubg 17146 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
21203adant1 1006 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
2210resghm2 15868 . . 3  |-  ( ( F  e.  ( S 
GrpHom  U )  /\  X  e.  (SubGrp `  T )
)  ->  F  e.  ( S  GrpHom  T ) )
2318, 21, 22syl2anc 661 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2451 . . . . . 6  |-  ( .s
`  U )  =  ( .s `  U
)
254, 6, 1, 2, 24lmhmlin 17224 . . . . 5  |-  ( ( F  e.  ( S LMHom 
U )  /\  x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) )  -> 
( F `  (
x ( .s `  S ) y ) )  =  ( x ( .s `  U
) ( F `  y ) ) )
26253expb 1189 . . . 4  |-  ( ( F  e.  ( S LMHom 
U )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( .s `  S
) y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
27263ad2antl1 1150 . . 3  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( .s `  S
) y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
28 simpl3 993 . . . 4  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  X  e.  L )
2910, 3ressvsca 14421 . . . . 5  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3029oveqd 6209 . . . 4  |-  ( X  e.  L  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
3128, 30syl 16 . . 3  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
3227, 31eqtr4d 2495 . 2  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( .s `  S
) y ) )  =  ( x ( .s `  T ) ( F `  y
) ) )
331, 2, 3, 4, 5, 6, 8, 9, 16, 23, 32islmhmd 17228 1  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5518  (class class class)co 6192   Basecbs 14278   ↾s cress 14279  Scalarcsca 14345   .scvsca 14346  SubGrpcsubg 15779    GrpHom cghm 15848   LModclmod 17056   LSubSpclss 17121   LMHom clmhm 17208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-sca 14358  df-vsca 14359  df-0g 14484  df-mnd 15519  df-mhm 15568  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-subg 15782  df-ghm 15849  df-mgp 16699  df-ur 16711  df-rng 16755  df-lmod 17058  df-lss 17122  df-lmhm 17211
This theorem is referenced by:  reslmhm2b  17243
  Copyright terms: Public domain W3C validator