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Theorem 0lmhm 17896
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 2400 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2400 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 0lmhm.s . 2  |-  S  =  (Scalar `  M )
5 0lmhm.t . 2  |-  T  =  (Scalar `  N )
6 eqid 2400 . 2  |-  ( Base `  S )  =  (
Base `  S )
7 simp1 995 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  M  e.  LMod )
8 simp2 996 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  N  e.  LMod )
9 simp3 997 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  S  =  T )
109eqcomd 2408 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  T  =  S )
11 lmodgrp 17729 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
12 lmodgrp 17729 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Grp )
13 0lmhm.z . . . . 5  |-  .0.  =  ( 0g `  N )
1413, 10ghm 16495 . . . 4  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
1511, 12, 14syl2an 475 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
16153adant3 1015 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
17 simpl2 999 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  N  e.  LMod )
18 simprl 755 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  S ) )
19 simpl3 1000 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  S  =  T )
2019fveq2d 5807 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( Base `  S )  =  ( Base `  T
) )
2118, 20eleqtrd 2490 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  T ) )
22 eqid 2400 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
235, 3, 22, 13lmodvs0 17756 . . . 4  |-  ( ( N  e.  LMod  /\  x  e.  ( Base `  T
) )  ->  (
x ( .s `  N )  .0.  )  =  .0.  )
2417, 21, 23syl2anc 659 . . 3  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  N )  .0.  )  =  .0.  )
25 fvex 5813 . . . . . . 7  |-  ( 0g
`  N )  e. 
_V
2613, 25eqeltri 2484 . . . . . 6  |-  .0.  e.  _V
2726fvconst2 6061 . . . . 5  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2827oveq2d 6248 . . . 4  |-  ( y  e.  B  ->  (
x ( .s `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  ( x ( .s `  N
)  .0.  ) )
2928ad2antll 727 . . 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 998 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  M  e.  LMod )
31 simprr 756 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
y  e.  B )
321, 4, 2, 6lmodvscl 17739 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  S
)  /\  y  e.  B )  ->  (
x ( .s `  M ) y )  e.  B )
3330, 18, 31, 32syl3anc 1228 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) y )  e.  B )
3426fvconst2 6061 . . . 4  |-  ( ( x ( .s `  M ) y )  e.  B  ->  (
( B  X.  {  .0.  } ) `  (
x ( .s `  M ) y ) )  =  .0.  )
3533, 34syl 17 . . 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 2452 . 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 17895 1  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
)  e.  ( M LMHom 
N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   _Vcvv 3056   {csn 3969    X. cxp 4938   ` cfv 5523  (class class class)co 6232   Basecbs 14731  Scalarcsca 14802   .scvsca 14803   0gc0g 14944   Grpcgrp 16267    GrpHom cghm 16478   LModclmod 17722   LMHom clmhm 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-recs 6997  df-rdg 7031  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-plusg 14812  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-mhm 16180  df-grp 16271  df-ghm 16479  df-mgp 17352  df-ring 17410  df-lmod 17724  df-lmhm 17878
This theorem is referenced by:  0nmhm  21444  mendring  35469
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