Theorem idressidex0 18779

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 18767	. 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 17336	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
11	8, 10	syl 18	. . . . 5 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
12	7, 11	eqtr4id 2816	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐴)
13	6, 12	eleqtrrd 2865	. . 3 ⊢ (𝜑 → 0 ∈ 𝐶)
14	9, 1	ressbasss 17337	. . . . . . . 8 ⊢ (Base‘𝑆) ⊆ 𝐵
15	7, 14	eqsstri 3980	. . . . . . 7 ⊢ 𝐶 ⊆ 𝐵
16		ssralv 4003	. . . . . . 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 7424	. . . . . . 7 ⊢ (𝑒 = 0 → (𝑒 + 𝑥) = ( 0 + 𝑥))
21	20	eqeq1d 2764	. . . . . 6 ⊢ (𝑒 = 0 → ((𝑒 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
22	21	ovanraleqv 7441	. . . . 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 3570	. 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 3078  ∃wrex 3088   ⊆ wss 3902  ‘cfv 6537  (class class class)co 7417  Basecbs 17307   ↾_s cress 17328  +_gcplusg 17348  0_gc0g 17530

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-1cn 11186  ax-addcl 11188

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-nn 12262  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-0g 17532

This theorem is used by:  idressidex  18780  idressid  18781