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Theorem 0lmhm 15813
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 2296 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2296 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 0lmhm.s . 2  |-  S  =  (Scalar `  M )
5 0lmhm.t . 2  |-  T  =  (Scalar `  N )
6 eqid 2296 . 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 2301 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  T  =  S )
11 lmodgrp 15650 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
12 lmodgrp 15650 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Grp )
13 0lmhm.z . . . . 5  |-  .0.  =  ( 0g `  N )
1413, 10ghm 14713 . . . 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 5545 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( Base `  S )  =  ( Base `  T
) )
2118, 20eleqtrd 2372 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  T ) )
22 eqid 2296 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
235, 3, 22, 13lmodvs0 15680 . . . 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 5555 . . . . . . 7  |-  ( 0g
`  N )  e. 
_V
2613, 25eqeltri 2366 . . . . . 6  |-  .0.  e.  _V
2726fvconst2 5745 . . . . 5  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2827oveq2d 5890 . . . 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 15660 . . . . 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 5745 . . . 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 2339 . 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 15812 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 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   Grpcgrp 14378    GrpHom cghm 14696   LModclmod 15643   LMHom clmhm 15792
This theorem is referenced by:  0nmhm  18280  mendrng  27603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-mgp 15342  df-rng 15356  df-lmod 15645  df-lmhm 15795
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