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Theorem slmdacl 27986
Description: Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdacl.f  |-  F  =  (Scalar `  W )
slmdacl.k  |-  K  =  ( Base `  F
)
slmdacl.p  |-  .+  =  ( +g  `  F )
Assertion
Ref Expression
slmdacl  |-  ( ( W  e. SLMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )

Proof of Theorem slmdacl
StepHypRef Expression
1 slmdacl.f . . . 4  |-  F  =  (Scalar `  W )
21slmdsrg 27984 . . 3  |-  ( W  e. SLMod  ->  F  e. SRing )
3 srgmnd 17356 . . 3  |-  ( F  e. SRing  ->  F  e.  Mnd )
42, 3syl 16 . 2  |-  ( W  e. SLMod  ->  F  e.  Mnd )
5 slmdacl.k . . 3  |-  K  =  ( Base `  F
)
6 slmdacl.p . . 3  |-  .+  =  ( +g  `  F )
75, 6mndcl 16128 . 2  |-  ( ( F  e.  Mnd  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
84, 7syl3an1 1259 1  |-  ( ( W  e. SLMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784  Scalarcsca 14787   Mndcmnd 16118  SRingcsrg 17352  SLModcslmd 27977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-cmn 16999  df-srg 17353  df-slmd 27978
This theorem is referenced by: (None)
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