Theorem 0gisid 18853

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 7427	. . . . . 6 ⊢ (𝑒 = 𝑖 → (𝑒 + 𝑥) = (𝑖 + 𝑥))
3	2	eqeq1d 2762	. . . . 5 ⊢ (𝑒 = 𝑖 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑖 + 𝑥) = 𝑥))
4	3	ovanraleqv 7444	. . . 4 ⊢ (𝑒 = 𝑖 → (∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑖 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑖) = 𝑥)))
5	4	cbvrexvw 3241	. . 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 18850	. . . . . 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 2848	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑖 ∈ 𝐵 ↔ 0 ∈ 𝐵))
12		oveq1 7427	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑖 + 𝑥) = ( 0 + 𝑥))
13	12	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → ((𝑖 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
14	13	ovanraleqv 7444	. . . . . . . . . 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 2768	. . . . . . . 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 3164	. . 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  +_gcplusg 17433  0_gc0g 17615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-riota 7377  df-ov 7423  df-0g 17617

This theorem is used by:  idressidex0  18865  idressid  18867