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Theorem 0lmhm 16154
Description: The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
0lmhm.z  |-  .0.  =  ( 0g `  N )
0lmhm.b  |-  B  =  ( Base `  M
)
0lmhm.s  |-  S  =  (Scalar `  M )
0lmhm.t  |-  T  =  (Scalar `  N )
Assertion
Ref Expression
0lmhm  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
)  e.  ( M LMHom 
N ) )

Proof of Theorem 0lmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0lmhm.b . 2  |-  B  =  ( Base `  M
)
2 eqid 2443 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2443 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 0lmhm.s . 2  |-  S  =  (Scalar `  M )
5 0lmhm.t . 2  |-  T  =  (Scalar `  N )
6 eqid 2443 . 2  |-  ( Base `  S )  =  (
Base `  S )
7 simp1 958 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  M  e.  LMod )
8 simp2 959 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  N  e.  LMod )
9 simp3 960 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  S  =  T )
109eqcomd 2448 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  T  =  S )
11 lmodgrp 15995 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
12 lmodgrp 15995 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Grp )
13 0lmhm.z . . . . 5  |-  .0.  =  ( 0g `  N )
1413, 10ghm 15058 . . . 4  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
1511, 12, 14syl2an 465 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
16153adant3 978 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
17 simpl2 962 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  N  e.  LMod )
18 simprl 734 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  S ) )
19 simpl3 963 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  S  =  T )
2019fveq2d 5767 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( Base `  S )  =  ( Base `  T
) )
2118, 20eleqtrd 2519 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  T ) )
22 eqid 2443 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
235, 3, 22, 13lmodvs0 16022 . . . 4  |-  ( ( N  e.  LMod  /\  x  e.  ( Base `  T
) )  ->  (
x ( .s `  N )  .0.  )  =  .0.  )
2417, 21, 23syl2anc 644 . . 3  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  N )  .0.  )  =  .0.  )
25 fvex 5773 . . . . . . 7  |-  ( 0g
`  N )  e. 
_V
2613, 25eqeltri 2513 . . . . . 6  |-  .0.  e.  _V
2726fvconst2 5983 . . . . 5  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2827oveq2d 6133 . . . 4  |-  ( y  e.  B  ->  (
x ( .s `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  ( x ( .s `  N
)  .0.  ) )
2928ad2antll 711 . . 3  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  N ) ( ( B  X.  {  .0.  } ) `  y
) )  =  ( x ( .s `  N )  .0.  )
)
30 simpl1 961 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  M  e.  LMod )
31 simprr 735 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
y  e.  B )
321, 4, 2, 6lmodvscl 16005 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  S
)  /\  y  e.  B )  ->  (
x ( .s `  M ) y )  e.  B )
3330, 18, 31, 32syl3anc 1185 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) y )  e.  B )
3426fvconst2 5983 . . . 4  |-  ( ( x ( .s `  M ) y )  e.  B  ->  (
( B  X.  {  .0.  } ) `  (
x ( .s `  M ) y ) )  =  .0.  )
3533, 34syl 16 . . 3  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( ( B  X.  {  .0.  } ) `  ( x ( .s
`  M ) y ) )  =  .0.  )
3624, 29, 353eqtr4rd 2486 . 2  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( ( B  X.  {  .0.  } ) `  ( x ( .s
`  M ) y ) )  =  ( x ( .s `  N ) ( ( B  X.  {  .0.  } ) `  y ) ) )
371, 2, 3, 4, 5, 6, 7, 8, 10, 16, 36islmhmd 16153 1  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
)  e.  ( M LMHom 
N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   _Vcvv 2965   {csn 3843    X. cxp 4911   ` cfv 5489  (class class class)co 6117   Basecbs 13507  Scalarcsca 13570   .scvsca 13571   0gc0g 13761   Grpcgrp 14723    GrpHom cghm 15041   LModclmod 15988   LMHom clmhm 16133
This theorem is referenced by:  0nmhm  18827  mendrng  27589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  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 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-recs 6669  df-rdg 6704  df-er 6941  df-map 7056  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-ndx 13510  df-slot 13511  df-base 13512  df-sets 13513  df-plusg 13580  df-0g 13765  df-mnd 14728  df-mhm 14776  df-grp 14850  df-ghm 15042  df-mgp 15687  df-rng 15701  df-lmod 15990  df-lmhm 16136
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