Theorem idressid 18891

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 17445	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
6	2, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
7	1, 6	eleqtrd 2866	. 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 18877	. . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
12		ssralv 3982	. . . . . 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 6922	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
15	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
16	15, 2	ssexd 5290	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ V)
17	3, 9	ressplusg 17491	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → + = (+g‘𝑆))
18	16, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → + = (+g‘𝑆))
19	18	oveqd 7460	. . . . . . . 8 ⊢ (𝜑 → ( 0 + 𝑥) = ( 0 (+g‘𝑆)𝑥))
20	19	eqeq1d 2765	. . . . . . 7 ⊢ (𝜑 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 (+g‘𝑆)𝑥) = 𝑥))
21	18	oveqd 7460	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + 0 ) = (𝑥(+g‘𝑆) 0 ))
22	21	eqeq1d 2765	. . . . . . 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 3338	. . . . 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 2763	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
29		eqid 2763	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
30		eqid 2763	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
31	4, 9, 8, 10, 3, 2, 1, 28	idressidex0 18889	. . . 4 ⊢ (𝜑 → ∃𝑒 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
32	18	eqcomd 2769	. . . . . . . . 9 ⊢ (𝜑 → (+g‘𝑆) = + )
33	32	oveqd 7460	. . . . . . . 8 ⊢ (𝜑 → (𝑒(+g‘𝑆)𝑥) = (𝑒 + 𝑥))
34	33	eqeq1d 2765	. . . . . . 7 ⊢ (𝜑 → ((𝑒(+g‘𝑆)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
35	32	oveqd 7460	. . . . . . . 8 ⊢ (𝜑 → (𝑥(+g‘𝑆)𝑒) = (𝑥 + 𝑒))
36	35	eqeq1d 2765	. . . . . . 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 3189	. . . . 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 3190	. . . 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 18874	. 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 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ‘cfv 6558  (class class class)co 7443  Basecbs 17416   ↾_s cress 17437  +_gcplusg 17457  0_gc0g 17639

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-0g 17641

This theorem is used by:  ress0g  18983