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Theorem grpcld 18875
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 18869 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
71, 2, 3, 6syl3anc 1370 1 (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  Grpcgrp 18861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864
This theorem is referenced by:  dfgrp3  18965  xpsinv  18986  xpsgrpsub  18987  nmzsubg  19088  eqger  19101  lringuplu  20440  rnglidl1  21122  rngqiprngimfo  21149  rngqiprngfulem3  21161  cphpyth  25064  ghmquskerlem3  32971  evls1addd  33088  ply1degltlss  33108  q1pdir  33114  r1pcyc  33118  r1padd1  33119  r1plmhm  33121  evls1maprhm  33214  algextdeglem8  33235  grpcominv1  41549  rhmmpl  41588  mplmapghm  41591  evlsmaprhm  41605  evladdval  41610  selvadd  41623
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