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Theorem 0lmhm 17464
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 2462 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2462 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 0lmhm.s . 2  |-  S  =  (Scalar `  M )
5 0lmhm.t . 2  |-  T  =  (Scalar `  N )
6 eqid 2462 . 2  |-  ( Base `  S )  =  (
Base `  S )
7 simp1 991 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  M  e.  LMod )
8 simp2 992 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  N  e.  LMod )
9 simp3 993 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  S  =  T )
109eqcomd 2470 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  T  =  S )
11 lmodgrp 17297 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
12 lmodgrp 17297 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Grp )
13 0lmhm.z . . . . 5  |-  .0.  =  ( 0g `  N )
1413, 10ghm 16071 . . . 4  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
1511, 12, 14syl2an 477 . . 3  |-  ( ( M  e.  LMod  /\  N  e.  LMod )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
16153adant3 1011 . 2  |-  ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
)  e.  ( M 
GrpHom  N ) )
17 simpl2 995 . . . 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 996 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  S  =  T )
2019fveq2d 5863 . . . . 5  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( Base `  S )  =  ( Base `  T
) )
2118, 20eleqtrd 2552 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  ->  x  e.  ( Base `  T ) )
22 eqid 2462 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
235, 3, 22, 13lmodvs0 17324 . . . 4  |-  ( ( N  e.  LMod  /\  x  e.  ( Base `  T
) )  ->  (
x ( .s `  N )  .0.  )  =  .0.  )
2417, 21, 23syl2anc 661 . . 3  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  N )  .0.  )  =  .0.  )
25 fvex 5869 . . . . . . 7  |-  ( 0g
`  N )  e. 
_V
2613, 25eqeltri 2546 . . . . . 6  |-  .0.  e.  _V
2726fvconst2 6109 . . . . 5  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2827oveq2d 6293 . . . 4  |-  ( y  e.  B  ->  (
x ( .s `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  ( x ( .s `  N
)  .0.  ) )
2928ad2antll 728 . . 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 994 . . . . 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 17307 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  S
)  /\  y  e.  B )  ->  (
x ( .s `  M ) y )  e.  B )
3330, 18, 31, 32syl3anc 1223 . . . 4  |-  ( ( ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  B ) )  -> 
( x ( .s
`  M ) y )  e.  B )
3426fvconst2 6109 . . . 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 2514 . 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 17463 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3108   {csn 4022    X. cxp 4992   ` cfv 5581  (class class class)co 6277   Basecbs 14481  Scalarcsca 14549   .scvsca 14550   0gc0g 14686   Grpcgrp 15718    GrpHom cghm 16054   LModclmod 17290   LMHom clmhm 17443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-plusg 14559  df-0g 14688  df-mnd 15723  df-mhm 15772  df-grp 15853  df-ghm 16055  df-mgp 16927  df-rng 16983  df-lmod 17292  df-lmhm 17446
This theorem is referenced by:  0nmhm  20992  mendrng  30737
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