Theorem idressid 18781

Description: The restriction of a structure with an identity element to a subset containing the identity element has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 12-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
idressid	⊢ (𝜑 → (0g‘𝑆) = 0 )

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

Proof of Theorem idressid

Step	Hyp	Ref	Expression
1		idressidex.0	. . 3 ⊢ (𝜑 → 0 ∈ 𝐴)
2		idressidex.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		idressidex.s	. . . . 5 ⊢ 𝑆 = (𝐺 ↾s 𝐴)
4		idressidex.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	ressbas2 17336	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
6	2, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
7	1, 6	eleqtrd 2864	. 2 ⊢ (𝜑 → 0 ∈ (Base‘𝑆))
8		idressidex.o	. . . 4 ⊢ 0 = (0g‘𝐺)
9		idressidex.p	. . . 4 ⊢ + = (+g‘𝐺)
10		idressidex.e	. . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
11	4, 8, 9, 10	0gisid 18767	. . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
12		ssralv 4003	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) → ∀𝑥 ∈ 𝐴 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
13	2, 12	syl 18	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) → ∀𝑥 ∈ 𝐴 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
14	4	fvexi 6896	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
15	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
16	15, 2	ssexd 5293	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ V)
17	3, 9	ressplusg 17382	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → + = (+g‘𝑆))
18	16, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → + = (+g‘𝑆))
19	18	oveqd 7434	. . . . . . . 8 ⊢ (𝜑 → ( 0 + 𝑥) = ( 0 (+g‘𝑆)𝑥))
20	19	eqeq1d 2764	. . . . . . 7 ⊢ (𝜑 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 (+g‘𝑆)𝑥) = 𝑥))
21	18	oveqd 7434	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + 0 ) = (𝑥(+g‘𝑆) 0 ))
22	21	eqeq1d 2764	. . . . . . 7 ⊢ (𝜑 → ((𝑥 + 0 ) = 𝑥 ↔ (𝑥(+g‘𝑆) 0 ) = 𝑥))
23	20, 22	anbi12d 644	. . . . . 6 ⊢ (𝜑 → ((( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ↔ (( 0 (+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆) 0 ) = 𝑥)))
24	6, 23	raleqbidv 3336	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑆)(( 0 (+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆) 0 ) = 𝑥)))
25	13, 24	sylibd 242	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) → ∀𝑥 ∈ (Base‘𝑆)(( 0 (+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆) 0 ) = 𝑥)))
26	25	adantld 496	. . 3 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) → ∀𝑥 ∈ (Base‘𝑆)(( 0 (+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆) 0 ) = 𝑥)))
27	11, 26	mpd 16	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑆)(( 0 (+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆) 0 ) = 𝑥))
28		eqid 2762	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
29		eqid 2762	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
30		eqid 2762	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
31	4, 9, 8, 10, 3, 2, 1, 28	idressidex0 18779	. . . 4 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
32	18	eqcomd 2768	. . . . . . . . 9 ⊢ (𝜑 → (+g‘𝑆) = + )
33	32	oveqd 7434	. . . . . . . 8 ⊢ (𝜑 → (𝑒(+g‘𝑆)𝑥) = (𝑒 + 𝑥))
34	33	eqeq1d 2764	. . . . . . 7 ⊢ (𝜑 → ((𝑒(+g‘𝑆)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
35	32	oveqd 7434	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+g‘𝑆)𝑒) = (𝑥 + 𝑒))
36	35	eqeq1d 2764	. . . . . . 7 ⊢ (𝜑 → ((𝑥(+g‘𝑆)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
37	34, 36	anbi12d 644	. . . . . 6 ⊢ (𝜑 → (((𝑒(+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
38	37	ralbidv 3187	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝑆)((𝑒(+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆)𝑒) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
39	38	rexbidv 3188	. . . 4 ⊢ (𝜑 → (∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒(+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆)𝑒) = 𝑥) ↔ ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
40	31, 39	mpbird 260	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒(+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆)𝑒) = 𝑥))
41	28, 29, 30, 40	ismgmid 18764	. 2 ⊢ (𝜑 → (( 0 ∈ (Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)(( 0 (+g‘𝑆)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑆) 0 ) = 𝑥)) ↔ (0g‘𝑆) = 0 ))
42	7, 27, 41	mpbi2and 725	1 ⊢ (𝜑 → (0g‘𝑆) = 0 )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ⊆ 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-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

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-nel 3064  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-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-0g 17532

This theorem is used by:  ress0g  18873