Theorem idressid 18762

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 17323	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
6	2, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
7	1, 6	eleqtrd 2868	. 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 18751	. . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
12		ssralv 4009	. . . . . 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 6902	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
15	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
16	15, 2	ssexd 5300	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ V)
17	3, 9	ressplusg 17369	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → + = (+g‘𝑆))
18	16, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → + = (+g‘𝑆))
19	18	oveqd 7440	. . . . . . . 8 ⊢ (𝜑 → ( 0 + 𝑥) = ( 0 (+g‘𝑆)𝑥))
20	19	eqeq1d 2768	. . . . . . 7 ⊢ (𝜑 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 (+g‘𝑆)𝑥) = 𝑥))
21	18	oveqd 7440	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + 0 ) = (𝑥(+g‘𝑆) 0 ))
22	21	eqeq1d 2768	. . . . . . 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 3341	. . . . 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 2766	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
29		eqid 2766	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
30		eqid 2766	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
31	4, 9, 8, 10, 3, 2, 1, 28	idressidex0 18760	. . . 4 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
32	18	eqcomd 2772	. . . . . . . . 9 ⊢ (𝜑 → (+g‘𝑆) = + )
33	32	oveqd 7440	. . . . . . . 8 ⊢ (𝜑 → (𝑒(+g‘𝑆)𝑥) = (𝑒 + 𝑥))
34	33	eqeq1d 2768	. . . . . . 7 ⊢ (𝜑 → ((𝑒(+g‘𝑆)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
35	32	oveqd 7440	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+g‘𝑆)𝑒) = (𝑥 + 𝑒))
36	35	eqeq1d 2768	. . . . . . 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 3191	. . . . 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 3192	. . . 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 18748	. 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 2146  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ⊆ 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-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

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-nel 3068  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-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-0g 17519

This theorem is used by:  ress0g  18849