Theorem grpidcld 33168

Description: The identity element of a group belongs to the group. (Contributed by Thierry Arnoux, 4-May-2026.)

Hypotheses

Ref	Expression
grpidcld.1	⊢ 𝐵 = (Base‘𝐺)
grpidcld.2	⊢ 0 = (0g‘𝐺)
grpidcld.3	⊢ (𝜑 → 𝐺 ∈ Grp)

Assertion

Ref	Expression
grpidcld	⊢ (𝜑 → 0 ∈ 𝐵)

Proof of Theorem grpidcld

Step	Hyp	Ref	Expression
1		grpidcld.3	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grpidcld.1	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		grpidcld.2	. . 3 ⊢ 0 = (0g‘𝐺)
4	2, 3	grpidcl 18979	. 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
5	1, 4	syl 17	1 ⊢ (𝜑 → 0 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1550   ∈ wcel 2132  ‘cfv 6506  Basecbs 17217  0_gc0g 17440  Grpcgrp 18947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-iota 6462  df-fun 6508  df-fv 6514  df-riota 7338  df-ov 7384  df-0g 17442  df-mgm 18646  df-sgrp 18725  df-mnd 18741  df-grp 18950

This theorem is referenced by:  drnglring  33632  dflringlem2  33635  mplasclco  33757  selvply1rhmlem2  33762