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Theorem reslmhm2b 15904
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 2358 . . 3  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2358 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2358 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
4 eqid 2358 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2358 . . 3  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2358 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 15883 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantl 452 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
9 simpl1 958 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  T  e.  LMod )
10 simpl2 959 . . . 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 15810 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  U  e.  LMod )
149, 10, 13syl2anc 642 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  U  e.  LMod )
15 eqid 2358 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
1611, 15resssca 13374 . . . . 5  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
17163ad2ant2 977 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  (Scalar `  T )  =  (Scalar `  U ) )
184, 15lmhmsca 15880 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1917, 18sylan9req 2411 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  -> 
(Scalar `  U )  =  (Scalar `  S )
)
20 lmghm 15881 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2112lsssubg 15807 . . . . . . 7  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
22213adant3 975 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  X  e.  (SubGrp `  T )
)
23 simp3 957 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ran  F 
C_  X )
2411resghm2b 14794 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2522, 23, 24syl2anc 642 . . . . 5  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2625biimpa 470 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S  GrpHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
2720, 26sylan2 460 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
28 eqid 2358 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
294, 6, 1, 2, 28lmhmlin 15885 . . . . . 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 1152 . . . . 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 694 . . . 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 995 . . . . 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 13375 . . . . . 6  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3433oveqd 5959 . . . . 5  |-  ( X  e.  L  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
3532, 34syl 15 . . . 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 2390 . . 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 15889 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S LMHom  U ) )
38 simpr 447 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  U ) )
39 simpl1 958 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  T  e.  LMod )
40 simpl2 959 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  X  e.  L )
4111, 12reslmhm2 15903 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
4238, 39, 40, 41syl3anc 1182 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  T ) )
4337, 42impbida 805 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    C_ wss 3228   ran crn 4769   ` cfv 5334  (class class class)co 5942   Basecbs 13239   ↾s cress 13240  Scalarcsca 13302   .scvsca 13303  SubGrpcsubg 14708    GrpHom cghm 14773   LModclmod 15720   LSubSpclss 15782   LMHom clmhm 15869
This theorem is referenced by:  pj1lmhm2  15947  frlmsplit2  26566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-sca 13315  df-vsca 13316  df-0g 13497  df-mnd 14460  df-mhm 14508  df-submnd 14509  df-grp 14582  df-minusg 14583  df-sbg 14584  df-subg 14711  df-ghm 14774  df-mgp 15419  df-rng 15433  df-ur 15435  df-lmod 15722  df-lss 15783  df-lmhm 15872
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