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Theorem slmdacl 26363
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 26361 . . 3  |-  ( W  e. SLMod  ->  F  e. SRing )
3 srgmnd 16725 . . 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 15531 . 2  |-  ( ( F  e.  Mnd  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
84, 7syl3an1 1252 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 965    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   Basecbs 14285   +g cplusg 14349  Scalarcsca 14352   Mndcmnd 15520  SRingcsrg 16721  SLModcslmd 26354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522  ax-pow 4571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196  df-mnd 15526  df-cmn 16392  df-srg 16722  df-slmd 26355
This theorem is referenced by: (None)
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