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Theorem reslmhm2b 16093
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 2412 . . 3  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2412 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2412 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
4 eqid 2412 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2412 . . 3  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2412 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 16072 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
87adantl 453 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  S  e.  LMod )
9 simpl1 960 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  T  e.  LMod )
10 simpl2 961 . . . 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 15999 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  U  e.  LMod )
149, 10, 13syl2anc 643 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  U  e.  LMod )
15 eqid 2412 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
1611, 15resssca 13567 . . . . 5  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
17163ad2ant2 979 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  (Scalar `  T )  =  (Scalar `  U ) )
184, 15lmhmsca 16069 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1917, 18sylan9req 2465 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  -> 
(Scalar `  U )  =  (Scalar `  S )
)
20 lmghm 16070 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2112lsssubg 15996 . . . . . . 7  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
22213adant3 977 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  X  e.  (SubGrp `  T )
)
23 simp3 959 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ran  F 
C_  X )
2411resghm2b 14987 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2522, 23, 24syl2anc 643 . . . . 5  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
2625biimpa 471 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S  GrpHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
2720, 26sylan2 461 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S  GrpHom  U ) )
28 eqid 2412 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
294, 6, 1, 2, 28lmhmlin 16074 . . . . . 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 1154 . . . . 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 695 . . . 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 997 . . . . 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 13568 . . . . . 6  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3433oveqd 6065 . . . . 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 2444 . . 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 16078 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  F  e.  ( S LMHom  U ) )
38 simpr 448 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  U ) )
39 simpl1 960 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  T  e.  LMod )
40 simpl2 961 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  X  e.  L )
4111, 12reslmhm2 16092 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
4238, 39, 40, 41syl3anc 1184 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  T ) )
4337, 42impbida 806 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3288   ran crn 4846   ` cfv 5421  (class class class)co 6048   Basecbs 13432   ↾s cress 13433  Scalarcsca 13495   .scvsca 13496  SubGrpcsubg 14901    GrpHom cghm 14966   LModclmod 15913   LSubSpclss 15971   LMHom clmhm 16058
This theorem is referenced by:  pj1lmhm2  16136  frlmsplit2  27119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-sca 13508  df-vsca 13509  df-0g 13690  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-ghm 14967  df-mgp 15612  df-rng 15626  df-ur 15628  df-lmod 15915  df-lss 15972  df-lmhm 16061
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