Theorem grpidcld 33224

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 19017	. 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
5	1, 4	syl 17	1 ⊢ (𝜑 → 0 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  ‘cfv 6521  Basecbs 17255  0_gc0g 17478  Grpcgrp 18985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-riota 7353  df-ov 7399  df-0g 17480  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-grp 18988

This theorem is referenced by:  drnglring  33691  dflringlem2  33694  mplasclco  33815  selvply1rhmlem2  33820