Theorem idressidex 18866

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 2760	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	1, 2, 3, 4, 5, 6, 7, 8	idressidex0 18865	. 2 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
10	5, 1	ressbas2 17421	. . 3 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
11		id 23	. . . 4 ⊢ (𝐴 = (Base‘𝑆) → 𝐴 = (Base‘𝑆))
12		raleq 3316	. . . 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 3335	. . 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  Basecbs 17392   ↾_s cress 17413  +_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-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-1cn 11261  ax-addcl 11263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-nn 12339  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-0g 17617

This theorem is used by: (None)