Theorem idressidex0 18760

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 18751	. 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 17323	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
11	8, 10	syl 18	. . . . 5 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
12	7, 11	eqtr4id 2820	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐴)
13	6, 12	eleqtrrd 2869	. . 3 ⊢ (𝜑 → 0 ∈ 𝐶)
14	9, 1	ressbasss 17324	. . . . . . . 8 ⊢ (Base‘𝑆) ⊆ 𝐵
15	7, 14	eqsstri 3986	. . . . . . 7 ⊢ 𝐶 ⊆ 𝐵
16		ssralv 4009	. . . . . . 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 7430	. . . . . . 7 ⊢ (𝑒 = 0 → (𝑒 + 𝑥) = ( 0 + 𝑥))
21	20	eqeq1d 2768	. . . . . 6 ⊢ (𝑒 = 0 → ((𝑒 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
22	21	ovanraleqv 7447	. . . . 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 3575	. 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 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:  idressidex  18761  idressid  18762