Theorem idressidex 18890

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 2763	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	1, 2, 3, 4, 5, 6, 7, 8	idressidex0 18889	. 2 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
10	5, 1	ressbas2 17445	. . 3 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
11		id 23	. . . 4 ⊢ (𝐴 = (Base‘𝑆) → 𝐴 = (Base‘𝑆))
12		raleq 3320	. . . 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 3339	. . 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 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ‘cfv 6558  (class class class)co 7443  Basecbs 17416   ↾_s cress 17437  +_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-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-1cn 11285  ax-addcl 11287

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 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-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-nn 12363  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-0g 17641

This theorem is used by: (None)