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Theorem grpcld 18904
Description: Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
Hypotheses
Ref Expression
grpcld.b 𝐵 = (Base‘𝐺)
grpcld.p + = (+g𝐺)
grpcld.r (𝜑𝐺 ∈ Grp)
grpcld.x (𝜑𝑋𝐵)
grpcld.y (𝜑𝑌𝐵)
Assertion
Ref Expression
grpcld (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)

Proof of Theorem grpcld
StepHypRef Expression
1 grpcld.r . 2 (𝜑𝐺 ∈ Grp)
2 grpcld.x . 2 (𝜑𝑋𝐵)
3 grpcld.y . 2 (𝜑𝑌𝐵)
4 grpcld.b . . 3 𝐵 = (Base‘𝐺)
5 grpcld.p . . 3 + = (+g𝐺)
64, 5grpcl 18898 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
71, 2, 3, 6syl3anc 1369 1 (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  Grpcgrp 18890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893
This theorem is referenced by:  grpraddf1o  18970  dfgrp3  18995  xpsinv  19016  xpsgrpsub  19017  nmzsubg  19120  eqger  19133  conjnmz  19206  ghmquskerlem3  19237  lringuplu  20481  rnglidl1  21128  rngqiprngimfo  21191  rngqiprngfulem3  21203  psdmul  22090  evls1addd  22290  evls1maprhm  22295  cphpyth  25157  rlocaddval  32995  rloccring  32997  rlocf1  33000  ghmqusnsg  33144  ply1degltlss  33267  q1pdir  33273  r1pcyc  33277  r1padd1  33278  r1plmhm  33280  algextdeglem8  33392  aks6d1c1p3  41581  grpcominv1  41748  rhmmpl  41786  mplmapghm  41789  evlsmaprhm  41803  evladdval  41808  selvadd  41821
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