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Theorem reslmhm2 15810
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 2283 . 2  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2283 . 2  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2283 . 2  |-  ( .s
`  T )  =  ( .s `  T
)
4 eqid 2283 . 2  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2283 . 2  |-  (Scalar `  T )  =  (Scalar `  T )
6 eqid 2283 . 2  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 15790 . . 3  |-  ( F  e.  ( S LMHom  U
)  ->  S  e.  LMod )
873ad2ant1 976 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  S  e.  LMod )
9 simp2 956 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  T  e.  LMod )
10 reslmhm2.u . . . . 5  |-  U  =  ( Ts  X )
1110, 5resssca 13283 . . . 4  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
12113ad2ant3 978 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  U ) )
13 eqid 2283 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
144, 13lmhmsca 15787 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  (Scalar `  U
)  =  (Scalar `  S ) )
15143ad2ant1 976 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  U )  =  (Scalar `  S ) )
1612, 15eqtrd 2315 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  S ) )
17 lmghm 15788 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  F  e.  ( S  GrpHom  U ) )
18173ad2ant1 976 . . 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 15714 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
21203adant1 973 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
2210resghm2 14700 . . 3  |-  ( ( F  e.  ( S 
GrpHom  U )  /\  X  e.  (SubGrp `  T )
)  ->  F  e.  ( S  GrpHom  T ) )
2318, 21, 22syl2anc 642 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2283 . . . . . 6  |-  ( .s
`  U )  =  ( .s `  U
)
254, 6, 1, 2, 24lmhmlin 15792 . . . . 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 1152 . . . 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 1117 . . 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 960 . . . 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 13284 . . . . 5  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3029oveqd 5875 . . . 4  |-  ( X  e.  L  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
3128, 30syl 15 . . 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 2318 . 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 15796 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211   .scvsca 13212  SubGrpcsubg 14615    GrpHom cghm 14680   LModclmod 15627   LSubSpclss 15689   LMHom clmhm 15776
This theorem is referenced by:  reslmhm2b  15811
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-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-sca 13224  df-vsca 13225  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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