Theorem grpidcld 33325

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 19031	. 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
5	1, 4	syl 18	1 ⊢ (𝜑 → 0 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  ‘cfv 6536  Basecbs 17268  0_gc0g 17491  Grpcgrp 18999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-riota 7367  df-ov 7413  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002

This theorem is referenced by:  drnglring  33748  dflringlem2  33751  mplasclco  33872  selvply1rhmlem2  33877