Theorem idressidex0 18865

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 18853	. 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 17421	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆))
11	8, 10	syl 18	. . . . 5 ⊢ (𝜑 → 𝐴 = (Base‘𝑆))
12	7, 11	eqtr4id 2814	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐴)
13	6, 12	eleqtrrd 2863	. . 3 ⊢ (𝜑 → 0 ∈ 𝐶)
14	9, 1	ressbasss 17422	. . . . . . . 8 ⊢ (Base‘𝑆) ⊆ 𝐵
15	7, 14	eqsstri 3954	. . . . . . 7 ⊢ 𝐶 ⊆ 𝐵
16		ssralv 3977	. . . . . . 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 7427	. . . . . . 7 ⊢ (𝑒 = 0 → (𝑒 + 𝑥) = ( 0 + 𝑥))
21	20	eqeq1d 2762	. . . . . 6 ⊢ (𝑒 = 0 → ((𝑒 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
22	21	ovanraleqv 7444	. . . . 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 3567	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ 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-1cn 11261  ax-addcl 11263

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-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-nn 12339  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-0g 17617

This theorem is used by:  idressidex  18866  idressid  18867