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Theorem lspun0 13577
Description: The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
Hypotheses
Ref Expression
lspun0.v  |-  V  =  ( Base `  W
)
lspun0.o  |-  .0.  =  ( 0g `  W )
lspun0.n  |-  N  =  ( LSpan `  W )
lspun0.w  |-  ( ph  ->  W  e.  LMod )
lspun0.x  |-  ( ph  ->  X  C_  V )
Assertion
Ref Expression
lspun0  |-  ( ph  ->  ( N `  ( X  u.  {  .0.  } ) )  =  ( N `  X ) )

Proof of Theorem lspun0
StepHypRef Expression
1 lspun0.w . . 3  |-  ( ph  ->  W  e.  LMod )
2 lspun0.x . . 3  |-  ( ph  ->  X  C_  V )
3 lspun0.v . . . . . 6  |-  V  =  ( Base `  W
)
4 lspun0.o . . . . . 6  |-  .0.  =  ( 0g `  W )
53, 4lmod0vcl 13469 . . . . 5  |-  ( W  e.  LMod  ->  .0.  e.  V )
61, 5syl 14 . . . 4  |-  ( ph  ->  .0.  e.  V )
76snssd 3749 . . 3  |-  ( ph  ->  {  .0.  }  C_  V )
8 lspun0.n . . . 4  |-  N  =  ( LSpan `  W )
93, 8lspun 13554 . . 3  |-  ( ( W  e.  LMod  /\  X  C_  V  /\  {  .0.  } 
C_  V )  -> 
( N `  ( X  u.  {  .0.  } ) )  =  ( N `  ( ( N `  X )  u.  ( N `  {  .0.  } ) ) ) )
101, 2, 7, 9syl3anc 1248 . 2  |-  ( ph  ->  ( N `  ( X  u.  {  .0.  } ) )  =  ( N `  ( ( N `  X )  u.  ( N `  {  .0.  } ) ) ) )
114, 8lspsn0 13574 . . . . . . 7  |-  ( W  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
121, 11syl 14 . . . . . 6  |-  ( ph  ->  ( N `  {  .0.  } )  =  {  .0.  } )
1312uneq2d 3301 . . . . 5  |-  ( ph  ->  ( ( N `  X )  u.  ( N `  {  .0.  }
) )  =  ( ( N `  X
)  u.  {  .0.  } ) )
14 eqid 2187 . . . . . . . . 9  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
153, 14, 8lspcl 13543 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  C_  V )  ->  ( N `  X )  e.  ( LSubSp `  W )
)
161, 2, 15syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( N `  X
)  e.  ( LSubSp `  W ) )
174, 14lss0ss 13523 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( N `  X )  e.  ( LSubSp `  W )
)  ->  {  .0.  } 
C_  ( N `  X ) )
181, 16, 17syl2anc 411 . . . . . 6  |-  ( ph  ->  {  .0.  }  C_  ( N `  X ) )
19 ssequn2 3320 . . . . . 6  |-  ( {  .0.  }  C_  ( N `  X )  <->  ( ( N `  X
)  u.  {  .0.  } )  =  ( N `
 X ) )
2018, 19sylib 122 . . . . 5  |-  ( ph  ->  ( ( N `  X )  u.  {  .0.  } )  =  ( N `  X ) )
2113, 20eqtrd 2220 . . . 4  |-  ( ph  ->  ( ( N `  X )  u.  ( N `  {  .0.  }
) )  =  ( N `  X ) )
2221fveq2d 5531 . . 3  |-  ( ph  ->  ( N `  (
( N `  X
)  u.  ( N `
 {  .0.  }
) ) )  =  ( N `  ( N `  X )
) )
233, 8lspidm 13553 . . . 4  |-  ( ( W  e.  LMod  /\  X  C_  V )  ->  ( N `  ( N `  X ) )  =  ( N `  X
) )
241, 2, 23syl2anc 411 . . 3  |-  ( ph  ->  ( N `  ( N `  X )
)  =  ( N `
 X ) )
2522, 24eqtrd 2220 . 2  |-  ( ph  ->  ( N `  (
( N `  X
)  u.  ( N `
 {  .0.  }
) ) )  =  ( N `  X
) )
2610, 25eqtrd 2220 1  |-  ( ph  ->  ( N `  ( X  u.  {  .0.  } ) )  =  ( N `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158    u. cun 3139    C_ wss 3141   {csn 3604   ` cfv 5228   Basecbs 12475   0gc0g 12722   LModclmod 13439   LSubSpclss 13504   LSpanclspn 13538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-5 8994  df-6 8995  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-sca 12566  df-vsca 12567  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899  df-minusg 12900  df-sbg 12901  df-mgp 13163  df-ur 13197  df-ring 13235  df-lmod 13441  df-lssm 13505  df-lsp 13539
This theorem is referenced by: (None)
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