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Theorem reslmhm2 16129
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 2436 . 2  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2436 . 2  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2436 . 2  |-  ( .s
`  T )  =  ( .s `  T
)
4 eqid 2436 . 2  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2436 . 2  |-  (Scalar `  T )  =  (Scalar `  T )
6 eqid 2436 . 2  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 16109 . . 3  |-  ( F  e.  ( S LMHom  U
)  ->  S  e.  LMod )
873ad2ant1 978 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  S  e.  LMod )
9 simp2 958 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  T  e.  LMod )
10 reslmhm2.u . . . . 5  |-  U  =  ( Ts  X )
1110, 5resssca 13604 . . . 4  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
12113ad2ant3 980 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  U ) )
13 eqid 2436 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
144, 13lmhmsca 16106 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  (Scalar `  U
)  =  (Scalar `  S ) )
15143ad2ant1 978 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  U )  =  (Scalar `  S ) )
1612, 15eqtrd 2468 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  S ) )
17 lmghm 16107 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  F  e.  ( S  GrpHom  U ) )
18173ad2ant1 978 . . 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 16033 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
21203adant1 975 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
2210resghm2 15023 . . 3  |-  ( ( F  e.  ( S 
GrpHom  U )  /\  X  e.  (SubGrp `  T )
)  ->  F  e.  ( S  GrpHom  T ) )
2318, 21, 22syl2anc 643 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2436 . . . . . 6  |-  ( .s
`  U )  =  ( .s `  U
)
254, 6, 1, 2, 24lmhmlin 16111 . . . . 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 1154 . . . 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 1119 . . 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 962 . . . 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 13605 . . . . 5  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3029oveqd 6098 . . . 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 2471 . 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 16115 1  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470  Scalarcsca 13532   .scvsca 13533  SubGrpcsubg 14938    GrpHom cghm 15003   LModclmod 15950   LSubSpclss 16008   LMHom clmhm 16095
This theorem is referenced by:  reslmhm2b  16130
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-3 10059  df-4 10060  df-5 10061  df-6 10062  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-sca 13545  df-vsca 13546  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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