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Theorem srgacl 16757
Description: Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
srgacl.b  |-  B  =  ( Base `  R
)
srgacl.p  |-  .+  =  ( +g  `  R )
Assertion
Ref Expression
srgacl  |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )

Proof of Theorem srgacl
StepHypRef Expression
1 srgmnd 16743 . 2  |-  ( R  e. SRing  ->  R  e.  Mnd )
2 srgacl.b . . 3  |-  B  =  ( Base `  R
)
3 srgacl.p . . 3  |-  .+  =  ( +g  `  R )
42, 3mndcl 15543 . 2  |-  ( ( R  e.  Mnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
51, 4syl3an1 1252 1  |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   Basecbs 14296   +g cplusg 14361   Mndcmnd 15532  SRingcsrg 16739
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 1955  ax-ext 2432  ax-nul 4532  ax-pow 4581
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-mnd 15538  df-cmn 16404  df-srg 16740
This theorem is referenced by: (None)
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