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Theorem reslmhm2b 17135
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 2443 . . 3  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2443 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2443 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
4 eqid 2443 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2443 . . 3  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2443 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 17114 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantl 466 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
9 simpl1 991 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  T  e.  LMod )
10 simpl2 992 . . . 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 17041 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  U  e.  LMod )
149, 10, 13syl2anc 661 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  U  e.  LMod )
15 eqid 2443 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
1611, 15resssca 14316 . . . . 5  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
17163ad2ant2 1010 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  (Scalar `  T )  =  (Scalar `  U ) )
184, 15lmhmsca 17111 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1917, 18sylan9req 2496 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  -> 
(Scalar `  U )  =  (Scalar `  S )
)
20 lmghm 17112 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2112lsssubg 17038 . . . . . . 7  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
22213adant3 1008 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  X  e.  (SubGrp `  T )
)
23 simp3 990 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ran  F 
C_  X )
2411resghm2b 15765 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2522, 23, 24syl2anc 661 . . . . 5  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2625biimpa 484 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S  GrpHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
2720, 26sylan2 474 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
28 eqid 2443 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
294, 6, 1, 2, 28lmhmlin 17116 . . . . . 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 1188 . . . . 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 713 . . . 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 1028 . . . . 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 14317 . . . . . 6  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3433oveqd 6108 . . . . 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 2475 . . 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 17120 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S LMHom  U ) )
38 simpr 461 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  U ) )
39 simpl1 991 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  T  e.  LMod )
40 simpl2 992 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  X  e.  L )
4111, 12reslmhm2 17134 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
4238, 39, 40, 41syl3anc 1218 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  T ) )
4337, 42impbida 828 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3328   ran crn 4841   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾s cress 14175  Scalarcsca 14241   .scvsca 14242  SubGrpcsubg 15675    GrpHom cghm 15744   LModclmod 16948   LSubSpclss 17013   LMHom clmhm 17100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-sca 14254  df-vsca 14255  df-0g 14380  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-ghm 15745  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lss 17014  df-lmhm 17103
This theorem is referenced by:  pj1lmhm2  17182  frlmsplit2  18197
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