Theorem idressid 18867

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 17421	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
6	2, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
7	1, 6	eleqtrd 2862	. 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 18853	. . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
12		ssralv 3977	. . . . . 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 6899	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
15	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
16	15, 2	ssexd 5285	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ V)
17	3, 9	ressplusg 17467	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → + = (+g‘𝑆))
18	16, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → + = (+g‘𝑆))
19	18	oveqd 7437	. . . . . . . 8 ⊢ (𝜑 → ( 0 + 𝑥) = ( 0 (+g‘𝑆)𝑥))
20	19	eqeq1d 2762	. . . . . . 7 ⊢ (𝜑 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 (+g‘𝑆)𝑥) = 𝑥))
21	18	oveqd 7437	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + 0 ) = (𝑥(+g‘𝑆) 0 ))
22	21	eqeq1d 2762	. . . . . . 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 3334	. . . . 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 2760	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
29		eqid 2760	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
30		eqid 2760	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
31	4, 9, 8, 10, 3, 2, 1, 28	idressidex0 18865	. . . 4 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
32	18	eqcomd 2766	. . . . . . . . 9 ⊢ (𝜑 → (+g‘𝑆) = + )
33	32	oveqd 7437	. . . . . . . 8 ⊢ (𝜑 → (𝑒(+g‘𝑆)𝑥) = (𝑒 + 𝑥))
34	33	eqeq1d 2762	. . . . . . 7 ⊢ (𝜑 → ((𝑒(+g‘𝑆)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
35	32	oveqd 7437	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+g‘𝑆)𝑒) = (𝑥 + 𝑒))
36	35	eqeq1d 2762	. . . . . . 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 3185	. . . . 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 3186	. . . 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 18850	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ 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-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

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-nel 3062  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-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-0g 17617

This theorem is used by:  ress0g  18959