Theorem 0gisid 18877

Description: In a structure with an identity element, the group identity element is an identity element of the structure. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 11-Aug-2026.)

Hypotheses

Ref	Expression
ismgmid.b	⊢ 𝐵 = (Base‘𝐺)
ismgmid.o	⊢ 0 = (0g‘𝐺)
ismgmid.p	⊢ + = (+g‘𝐺)
mgmidcl.e	⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

Assertion

Ref	Expression
0gisid	⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))

Distinct variable groups:   𝑥,𝑒, +   0 ,𝑒,𝑥   𝐵,𝑒,𝑥   𝑒,𝐺,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑒)

Proof of Theorem 0gisid

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgmidcl.e	. 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
2		oveq1 7450	. . . . . 6 ⊢ (𝑒 = 𝑖 → (𝑒 + 𝑥) = (𝑖 + 𝑥))
3	2	eqeq1d 2765	. . . . 5 ⊢ (𝑒 = 𝑖 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑖 + 𝑥) = 𝑥))
4	3	ovanraleqv 7467	. . . 4 ⊢ (𝑒 = 𝑖 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥)))
5	4	cbvrexvw 3245	. . 3 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∃𝑖 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥))
6		ismgmid.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
7		ismgmid.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8		ismgmid.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	6, 7, 8, 1	ismgmid 18874	. . . . . 6 ⊢ (𝜑 → ((𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥)) ↔ 0 = 𝑖))
10	9	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥))) → 0 = 𝑖)
11		eleq1 2852	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑖 ∈ 𝐵 ↔ 0 ∈ 𝐵))
12		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑖 + 𝑥) = ( 0 + 𝑥))
13	12	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → ((𝑖 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
14	13	ovanraleqv 7467	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
15	11, 14	anbi12d 644	. . . . . . . . 9 ⊢ (𝑖 = 0 → ((𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥)) ↔ ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))))
16	15	eqcoms 2771	. . . . . . . 8 ⊢ ( 0 = 𝑖 → ((𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥)) ↔ ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))))
17	16	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 0 = 𝑖) → ((𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥)) ↔ ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))))
18	17	biimpd 232	. . . . . 6 ⊢ ((𝜑 ∧ 0 = 𝑖) → ((𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥)) → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))))
19	18	impancom 457	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥))) → ( 0 = 𝑖 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))))
20	10, 19	mpd 16	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥))) → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
21	20	rexlimdvaa 3168	. . 3 ⊢ (𝜑 → (∃𝑖 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥) → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))))
22	5, 21	biimtrid 245	. 2 ⊢ (𝜑 → (∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))))
23	1, 22	mpd 16	1 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  +_gcplusg 17457  0_gc0g 17639

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-riota 7400  df-ov 7446  df-0g 17641

This theorem is used by:  idressidex0  18889  idressid  18891