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Theorem reslmhm2b 16135
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
reslmhm2b  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ( F  e.  ( S LMHom  T )  <->  F  e.  ( S LMHom  U ) ) )

Proof of Theorem reslmhm2b
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2438 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2438 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
4 eqid 2438 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2438 . . 3  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2438 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 16114 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantl 454 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
9 simpl1 961 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  T  e.  LMod )
10 simpl2 962 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  X  e.  L )
11 reslmhm2.u . . . . 5  |-  U  =  ( Ts  X )
12 reslmhm2.l . . . . 5  |-  L  =  ( LSubSp `  T )
1311, 12lsslmod 16041 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  U  e.  LMod )
149, 10, 13syl2anc 644 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  U  e.  LMod )
15 eqid 2438 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
1611, 15resssca 13609 . . . . 5  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
17163ad2ant2 980 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  (Scalar `  T )  =  (Scalar `  U ) )
184, 15lmhmsca 16111 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1917, 18sylan9req 2491 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  -> 
(Scalar `  U )  =  (Scalar `  S )
)
20 lmghm 16112 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2112lsssubg 16038 . . . . . . 7  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
22213adant3 978 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  X  e.  (SubGrp `  T )
)
23 simp3 960 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ran  F 
C_  X )
2411resghm2b 15029 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2522, 23, 24syl2anc 644 . . . . 5  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2625biimpa 472 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S  GrpHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
2720, 26sylan2 462 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
28 eqid 2438 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
294, 6, 1, 2, 28lmhmlin 16116 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) )  -> 
( F `  (
x ( .s `  S ) y ) )  =  ( x ( .s `  T
) ( F `  y ) ) )
30293expb 1155 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( .s `  S
) y ) )  =  ( x ( .s `  T ) ( F `  y
) ) )
3130adantll 696 . . . 4  |-  ( ( ( ( T  e. 
LMod  /\  X  e.  L  /\  ran  F  C_  X
)  /\  F  e.  ( S LMHom  T ) )  /\  ( x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) ) )  ->  ( F `  ( x ( .s
`  S ) y ) )  =  ( x ( .s `  T ) ( F `
 y ) ) )
32 simpll2 998 . . . . 5  |-  ( ( ( ( T  e. 
LMod  /\  X  e.  L  /\  ran  F  C_  X
)  /\  F  e.  ( S LMHom  T ) )  /\  ( x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) ) )  ->  X  e.  L
)
3311, 28ressvsca 13610 . . . . . 6  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3433oveqd 6101 . . . . 5  |-  ( X  e.  L  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
3532, 34syl 16 . . . 4  |-  ( ( ( ( T  e. 
LMod  /\  X  e.  L  /\  ran  F  C_  X
)  /\  F  e.  ( S LMHom  T ) )  /\  ( x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) ) )  ->  ( x ( .s `  T ) ( F `  y
) )  =  ( x ( .s `  U ) ( F `
 y ) ) )
3631, 35eqtrd 2470 . . 3  |-  ( ( ( ( T  e. 
LMod  /\  X  e.  L  /\  ran  F  C_  X
)  /\  F  e.  ( S LMHom  T ) )  /\  ( x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) ) )  ->  ( F `  ( x ( .s
`  S ) y ) )  =  ( x ( .s `  U ) ( F `
 y ) ) )
371, 2, 3, 4, 5, 6, 8, 14, 19, 27, 36islmhmd 16120 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S LMHom  U ) )
38 simpr 449 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  U ) )
39 simpl1 961 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  T  e.  LMod )
40 simpl2 962 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  X  e.  L )
4111, 12reslmhm2 16134 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
4238, 39, 40, 41syl3anc 1185 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  T ) )
4337, 42impbida 807 1  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ( F  e.  ( S LMHom  T )  <->  F  e.  ( S LMHom  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   ran crn 4882   ` cfv 5457  (class class class)co 6084   Basecbs 13474   ↾s cress 13475  Scalarcsca 13537   .scvsca 13538  SubGrpcsubg 14943    GrpHom cghm 15008   LModclmod 15955   LSubSpclss 16013   LMHom clmhm 16100
This theorem is referenced by:  pj1lmhm2  16178  frlmsplit2  27234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-sca 13550  df-vsca 13551  df-0g 13732  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-subg 14946  df-ghm 15009  df-mgp 15654  df-rng 15668  df-ur 15670  df-lmod 15957  df-lss 16014  df-lmhm 16103
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