Theorem grpcld 18571

Description: Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)

Hypotheses

Ref	Expression
grpcld.b	⊢ 𝐵 = (Base‘𝐺)
grpcld.p	⊢ + = (+g‘𝐺)
grpcld.r	⊢ (𝜑 → 𝐺 ∈ Grp)
grpcld.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
grpcld.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
grpcld	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)

Proof of Theorem grpcld

Step	Hyp	Ref	Expression
1		grpcld.r	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grpcld.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		grpcld.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		grpcld.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
5		grpcld.p	. . 3 ⊢ + = (+g‘𝐺)
6	4, 5	grpcl 18566	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
7	1, 2, 3, 6	syl3anc 1369	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  +_gcplusg 16943  Grpcgrp 18558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-ov 7271  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-grp 18561

This theorem is referenced by:  cphpyth  24361