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Theorem reslmhm2 17812
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
reslmhm2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )

Proof of Theorem reslmhm2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2382 . 2  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2382 . 2  |-  ( .s
`  S )  =  ( .s `  S
)
3 eqid 2382 . 2  |-  ( .s
`  T )  =  ( .s `  T
)
4 eqid 2382 . 2  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2382 . 2  |-  (Scalar `  T )  =  (Scalar `  T )
6 eqid 2382 . 2  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
7 lmhmlmod1 17792 . . 3  |-  ( F  e.  ( S LMHom  U
)  ->  S  e.  LMod )
873ad2ant1 1015 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  S  e.  LMod )
9 simp2 995 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  T  e.  LMod )
10 reslmhm2.u . . . . 5  |-  U  =  ( Ts  X )
1110, 5resssca 14784 . . . 4  |-  ( X  e.  L  ->  (Scalar `  T )  =  (Scalar `  U ) )
12113ad2ant3 1017 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  U ) )
13 eqid 2382 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
144, 13lmhmsca 17789 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  (Scalar `  U
)  =  (Scalar `  S ) )
15143ad2ant1 1015 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  U )  =  (Scalar `  S ) )
1612, 15eqtrd 2423 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  (Scalar `  T )  =  (Scalar `  S ) )
17 lmghm 17790 . . . 4  |-  ( F  e.  ( S LMHom  U
)  ->  F  e.  ( S  GrpHom  U ) )
18173ad2ant1 1015 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S  GrpHom  U ) )
19 reslmhm2.l . . . . 5  |-  L  =  ( LSubSp `  T )
2019lsssubg 17716 . . . 4  |-  ( ( T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
21203adant1 1012 . . 3  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  X  e.  (SubGrp `  T )
)
2210resghm2 16401 . . 3  |-  ( ( F  e.  ( S 
GrpHom  U )  /\  X  e.  (SubGrp `  T )
)  ->  F  e.  ( S  GrpHom  T ) )
2318, 21, 22syl2anc 659 . 2  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2382 . . . . . 6  |-  ( .s
`  U )  =  ( .s `  U
)
254, 6, 1, 2, 24lmhmlin 17794 . . . . 5  |-  ( ( F  e.  ( S LMHom 
U )  /\  x  e.  ( Base `  (Scalar `  S ) )  /\  y  e.  ( Base `  S ) )  -> 
( F `  (
x ( .s `  S ) y ) )  =  ( x ( .s `  U
) ( F `  y ) ) )
26253expb 1195 . . . 4  |-  ( ( F  e.  ( S LMHom 
U )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( .s `  S
) y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
27263ad2antl1 1156 . . 3  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( .s `  S
) y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
28 simpl3 999 . . . 4  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  X  e.  L )
2910, 3ressvsca 14785 . . . . 5  |-  ( X  e.  L  ->  ( .s `  T )  =  ( .s `  U
) )
3029oveqd 6213 . . . 4  |-  ( X  e.  L  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
3128, 30syl 16 . . 3  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  (
x ( .s `  T ) ( F `
 y ) )  =  ( x ( .s `  U ) ( F `  y
) ) )
3227, 31eqtr4d 2426 . 2  |-  ( ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L )  /\  (
x  e.  ( Base `  (Scalar `  S )
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( .s `  S
) y ) )  =  ( x ( .s `  T ) ( F `  y
) ) )
331, 2, 3, 4, 5, 6, 8, 9, 16, 23, 32islmhmd 17798 1  |-  ( ( F  e.  ( S LMHom 
U )  /\  T  e.  LMod  /\  X  e.  L )  ->  F  e.  ( S LMHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635  Scalarcsca 14705   .scvsca 14706  SubGrpcsubg 16312    GrpHom cghm 16381   LModclmod 17625   LSubSpclss 17691   LMHom clmhm 17778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-sca 14718  df-vsca 14719  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-submnd 16084  df-grp 16174  df-minusg 16175  df-sbg 16176  df-subg 16315  df-ghm 16382  df-mgp 17255  df-ur 17267  df-ring 17313  df-lmod 17627  df-lss 17692  df-lmhm 17781
This theorem is referenced by:  reslmhm2b  17813
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