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Theorem reslmhm2b 17571
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 2467 . . 3  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2467 . . 3  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2467 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
4 eqid 2467 . . 3  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2467 . . 3  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2467 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 17550 . . . 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 999 . . . 4  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  ->  T  e.  LMod )
10 simpl2 1000 . . . 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 17477 . . . 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 2467 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
1611, 15resssca 14650 . . . . 5  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
17163ad2ant2 1018 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  (Scalar `  T )  =  (Scalar `  U ) )
184, 15lmhmsca 17547 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
1917, 18sylan9req 2529 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  T ) )  -> 
(Scalar `  U )  =  (Scalar `  S )
)
20 lmghm 17548 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
2112lsssubg 17474 . . . . . . 7  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
22213adant3 1016 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  X  e.  (SubGrp `  T )
)
23 simp3 998 . . . . . 6  |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ran  F 
C_  X )
2411resghm2b 16157 . . . . . 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 2467 . . . . . . 7  |-  ( .s
`  T )  =  ( .s `  T
)
294, 6, 1, 2, 28lmhmlin 17552 . . . . . 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 1197 . . . . 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 1036 . . . . 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 14651 . . . . . 6  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3433oveqd 6312 . . . . 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 2508 . . 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 17556 . 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 999 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  T  e.  LMod )
40 simpl2 1000 . . 3  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  X  e.  L )
4111, 12reslmhm2 17570 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
4238, 39, 40, 41syl3anc 1228 . 2  |-  ( ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  /\  F  e.  ( S LMHom  U ) )  ->  F  e.  ( S LMHom  T ) )
4337, 42impbida 830 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 973    = wceq 1379    e. wcel 1767    C_ wss 3481   ran crn 5006   ` cfv 5594  (class class class)co 6295   Basecbs 14507   ↾s cress 14508  Scalarcsca 14575   .scvsca 14576  SubGrpcsubg 16067    GrpHom cghm 16136   LModclmod 17383   LSubSpclss 17449   LMHom clmhm 17536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-sca 14588  df-vsca 14589  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-lmod 17385  df-lss 17450  df-lmhm 17539
This theorem is referenced by:  pj1lmhm2  17618  frlmsplit2  18672
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