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Theorem slmdsrg 28361
 Description: The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypothesis
Ref Expression
slmdsrg.1 Scalar
Assertion
Ref Expression
slmdsrg SLMod SRing

Proof of Theorem slmdsrg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . . 3
2 eqid 2429 . . 3
3 eqid 2429 . . 3
4 eqid 2429 . . 3
5 slmdsrg.1 . . 3 Scalar
6 eqid 2429 . . 3
7 eqid 2429 . . 3
8 eqid 2429 . . 3
9 eqid 2429 . . 3
10 eqid 2429 . . 3
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 28356 . 2 SLMod CMnd SRing
1211simp2bi 1021 1 SLMod SRing
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1870  wral 2782  cfv 5601  (class class class)co 6305  cbs 15084   cplusg 15152  cmulr 15153  Scalarcsca 15155  cvsca 15156  c0g 15297  CMndccmn 17365  cur 17670  SRingcsrg 17674  SLModcslmd 28354 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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556 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 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-slmd 28355 This theorem is referenced by:  slmdacl  28363  slmdmcl  28364  slmdsn0  28365  slmd0cl  28372  slmd1cl  28373  slmdvs0  28379  gsumvsca2  28385
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