Theorem idressidex 18761

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)

Assertion

Ref	Expression
idressidex	⊢ (𝜑 → ∃𝑒 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem idressidex

Step	Hyp	Ref	Expression
1		idressidex.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		idressidex.p	. . 3 ⊢ + = (+g‘𝐺)
3		idressidex.o	. . 3 ⊢ 0 = (0g‘𝐺)
4		idressidex.e	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
5		idressidex.s	. . 3 ⊢ 𝑆 = (𝐺 ↾s 𝐴)
6		idressidex.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7		idressidex.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝐴)
8		eqid 2766	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	1, 2, 3, 4, 5, 6, 7, 8	idressidex0 18760	. 2 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
10	5, 1	ressbas2 17323	. . 3 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
11		id 23	. . . 4 ⊢ (𝐴 = (Base‘𝑆) → 𝐴 = (Base‘𝑆))
12		raleq 3323	. . . 4 ⊢ (𝐴 = (Base‘𝑆) → (∀𝑥 ∈ 𝐴 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
13	11, 12	rexeqbidv 3342	. . 3 ⊢ (𝐴 = (Base‘𝑆) → (∃𝑒 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
14	6, 10, 13	3syl 19	. 2 ⊢ (𝜑 → (∃𝑒 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
15	9, 14	mpbird 260	1 ⊢ (𝜑 → ∃𝑒 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   ⊆ wss 3908  ‘cfv 6543  (class class class)co 7423  Basecbs 17294   ↾_s cress 17315  +_gcplusg 17335  0_gc0g 17517

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-1cn 11176  ax-addcl 11178

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-nn 12252  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-0g 17519

This theorem is used by: (None)