Theorem grpidcld 33122

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 18935	. 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
5	1, 4	syl 17	1 ⊢ (𝜑 → 0 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2115  ‘cfv 6488  Basecbs 17173  0_gc0g 17396  Grpcgrp 18903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-ral 3051  df-rex 3061  df-rmo 3341  df-reu 3342  df-rab 3389  df-v 3430  df-sbc 3727  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6444  df-fun 6490  df-fv 6496  df-riota 7316  df-ov 7362  df-0g 17398  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-grp 18906

This theorem is referenced by:  mplasclco  33703  selvply1rhmlem2  33708