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Theorem slmdacl 28520
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 28518 . . 3  |-  ( W  e. SLMod  ->  F  e. SRing )
3 srgmnd 17731 . . 3  |-  ( F  e. SRing  ->  F  e.  Mnd )
42, 3syl 17 . 2  |-  ( W  e. SLMod  ->  F  e.  Mnd )
5 slmdacl.k . . 3  |-  K  =  ( Base `  F
)
6 slmdacl.p . . 3  |-  .+  =  ( +g  `  F )
75, 6mndcl 16533 . 2  |-  ( ( F  e.  Mnd  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
84, 7syl3an1 1297 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 982    = wceq 1437    e. wcel 1868   ` cfv 5598  (class class class)co 6302   Basecbs 15109   +g cplusg 15178  Scalarcsca 15181   Mndcmnd 16523  SRingcsrg 17727  SLModcslmd 28511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-nul 4552
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-iota 5562  df-fv 5606  df-ov 6305  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-cmn 17420  df-srg 17728  df-slmd 28512
This theorem is referenced by: (None)
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