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Theorem 0lmhm 15797
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 2283 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2283 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 0lmhm.s . 2  |-  S  =  (Scalar `  M )
5 0lmhm.t . 2  |-  T  =  (Scalar `  N )
6 eqid 2283 . 2  |-  ( Base `  S )  =  (
Base `  S )
7 simp1 955 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  M  e.  LMod )
8 simp2 956 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  N  e.  LMod )
9 simp3 957 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  S  =  T )
109eqcomd 2288 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  T  =  S )
11 lmodgrp 15634 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
12 lmodgrp 15634 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Grp )
13 0lmhm.z . . . . 5  |-  .0.  =  ( 0g `  N )
1413, 10ghm 14697 . . . 4  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
1511, 12, 14syl2an 463 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
16153adant3 975 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
17 simpl2 959 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  N  e.  LMod )
18 simprl 732 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  S ) )
19 simpl3 960 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  S  =  T )
2019fveq2d 5529 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( Base `  S )  =  ( Base `  T
) )
2118, 20eleqtrd 2359 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  T ) )
22 eqid 2283 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
235, 3, 22, 13lmodvs0 15664 . . . 4  |-  ( ( N  e.  LMod  /\  x  e.  ( Base `  T
) )  ->  (
x ( .s `  N )  .0.  )  =  .0.  )
2417, 21, 23syl2anc 642 . . 3  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  N )  .0.  )  =  .0.  )
25 fvex 5539 . . . . . . 7  |-  ( 0g
`  N )  e. 
_V
2613, 25eqeltri 2353 . . . . . 6  |-  .0.  e.  _V
2726fvconst2 5729 . . . . 5  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2827oveq2d 5874 . . . 4  |-  ( y  e.  B  ->  (
x ( .s `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  ( x ( .s `  N
)  .0.  ) )
2928ad2antll 709 . . 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 958 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  M  e.  LMod )
31 simprr 733 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
y  e.  B )
321, 4, 2, 6lmodvscl 15644 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  S
)  /\  y  e.  B )  ->  (
x ( .s `  M ) y )  e.  B )
3330, 18, 31, 32syl3anc 1182 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) y )  e.  B )
3426fvconst2 5729 . . . 4  |-  ( ( x ( .s `  M ) y )  e.  B  ->  (
( B  X.  {  .0.  } ) `  (
x ( .s `  M ) y ) )  =  .0.  )
3533, 34syl 15 . . 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 2326 . 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 15796 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  0nmhm  18264  mendrng  27500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-mgp 15326  df-rng 15340  df-lmod 15629  df-lmhm 15779
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