Theorem idressidex0 18840

Description: The restriction of a structure with an identity element to a subset containing the identity element has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 11-Aug-2026.)

Hypotheses

Ref	Expression
idressidex.b	⊢ 𝐵 = (Base‘𝐺)
idressidex.p	⊢ + = (+g‘𝐺)
idressidex.o	⊢ 0 = (0g‘𝐺)
idressidex.e	⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
idressidex.s	⊢ 𝑆 = (𝐺 ↾s 𝐴)
idressidex.a	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
idressidex.0	⊢ (𝜑 → 0 ∈ 𝐴)
idressidex0.c	⊢ 𝐶 = (Base‘𝑆)

Assertion

Ref	Expression
idressidex0	⊢ (𝜑 → ∃𝑒 ∈ 𝐶 ∀𝑥 ∈ 𝐶 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

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

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥, 𝑒)   𝑆(𝑥, 𝑒)

Proof of Theorem idressidex0

Step	Hyp	Ref	Expression
1		idressidex.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		idressidex.o	. . 3 ⊢ 0 = (0g‘𝐺)
3		idressidex.p	. . 3 ⊢ + = (+g‘𝐺)
4		idressidex.e	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
5	1, 2, 3, 4	0gisid 18828	. 2 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
6		idressidex.0	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐴)
7		idressidex0.c	. . . . 5 ⊢ 𝐶 = (Base‘𝑆)
8		idressidex.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
9		idressidex.s	. . . . . . 7 ⊢ 𝑆 = (𝐺 ↾s 𝐴)
10	9, 1	ressbas2 17396	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
11	8, 10	syl 18	. . . . 5 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
12	7, 11	eqtr4id 2815	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐴)
13	6, 12	eleqtrrd 2864	. . 3 ⊢ (𝜑 → 0 ∈ 𝐶)
14	9, 1	ressbasss 17397	. . . . . . . 8 ⊢ (Base‘𝑆) ⊆ 𝐵
15	7, 14	eqsstri 3977	. . . . . . 7 ⊢ 𝐶 ⊆ 𝐵
16		ssralv 4000	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) → ∀𝑥 ∈ 𝐶 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
17	15, 16	mp1i 14	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) → ∀𝑥 ∈ 𝐶 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
18	17	adantld 496	. . . . 5 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) → ∀𝑥 ∈ 𝐶 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
19	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑒 = 0 ) → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) → ∀𝑥 ∈ 𝐶 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
20		oveq1 7419	. . . . . . 7 ⊢ (𝑒 = 0 → (𝑒 + 𝑥) = ( 0 + 𝑥))
21	20	eqeq1d 2763	. . . . . 6 ⊢ (𝑒 = 0 → ((𝑒 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
22	21	ovanraleqv 7436	. . . . 5 ⊢ (𝑒 = 0 → (∀𝑥 ∈ 𝐶 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐶 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
23	22	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑒 = 0 ) → (∀𝑥 ∈ 𝐶 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐶 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
24	19, 23	sylibrd 262	. . 3 ⊢ ((𝜑 ∧ 𝑒 = 0 ) → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) → ∀𝑥 ∈ 𝐶 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
25	13, 24	rspcimedv 3568	. 2 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) → ∃𝑒 ∈ 𝐶 ∀𝑥 ∈ 𝐶 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
26	5, 25	mpd 16	1 ⊢ (𝜑 → ∃𝑒 ∈ 𝐶 ∀𝑥 ∈ 𝐶 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ‘cfv 6531  (class class class)co 7412  Basecbs 17367   ↾_s cress 17388  +_gcplusg 17408  0_gc0g 17590

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-1cn 11239  ax-addcl 11241

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-nn 12317  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-0g 17592

This theorem is used by:  idressidex  18841  idressid  18842