Theorem grpinvssd 18927

Description: If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.)

Hypotheses

Ref	Expression
grpidssd.m	⊢ (𝜑 → 𝑀 ∈ Grp)
grpidssd.s	⊢ (𝜑 → 𝑆 ∈ Grp)
grpidssd.b	⊢ 𝐵 = (Base‘𝑆)
grpidssd.c	⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀))
grpidssd.o	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦))

Assertion

Ref	Expression
grpinvssd	⊢ (𝜑 → (𝑋 ∈ 𝐵 → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋)))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem grpinvssd

Step	Hyp	Ref	Expression
1		grpidssd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Grp)
2		grpidssd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2731	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
4	2, 3	grpinvcl 18897	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ 𝐵)
5	1, 4	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ 𝐵)
6		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
7		grpidssd.o	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦))
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦))
9		oveq1 7353	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦))
10		oveq1 7353	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦))
11	9, 10	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → ((𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦) ↔ (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦)))
12		oveq2 7354	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋))
13		oveq2 7354	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
14	12, 13	eqeq12d 2747	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦) ↔ (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋)))
15	11, 14	rspc2va 3589	. . . . 5 ⊢ (((((invg‘𝑆)‘𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦)) → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
16	5, 6, 8, 15	syl21anc 837	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
17		eqid 2731	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
18		eqid 2731	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	2, 17, 18, 3	grplinv 18899	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))
20	1, 19	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))
21		grpidssd.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Grp)
22		grpidssd.c	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀))
23	22	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑀))
24		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
25		eqid 2731	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
26		eqid 2731	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
27		eqid 2731	. . . . . . 7 ⊢ (invg‘𝑀) = (invg‘𝑀)
28	24, 25, 26, 27	grplinv 18899	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
29	21, 23, 28	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
30	21, 1, 2, 22, 7	grpidssd 18926	. . . . . 6 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑀) = (0g‘𝑆))
32	29, 31	eqtr2d 2767	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑆) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
33	16, 20, 32	3eqtrd 2770	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
34	21	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ Grp)
35	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐵 ⊆ (Base‘𝑀))
36	35, 5	sseldd 3935	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ (Base‘𝑀))
37	24, 27	grpinvcl 18897	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
38	21, 23, 37	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
39	24, 25	grprcan 18883	. . . 4 ⊢ ((𝑀 ∈ Grp ∧ (((invg‘𝑆)‘𝑋) ∈ (Base‘𝑀) ∧ ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀) ∧ 𝑋 ∈ (Base‘𝑀))) → ((((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) ↔ ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋)))
40	34, 36, 38, 23, 39	syl13anc 1374	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) ↔ ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋)))
41	33, 40	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋))
42	41	ex 412	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3902  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  0_gc0g 17340  Grpcgrp 18843  inv_gcminusg 18844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-riota 7303  df-ov 7349  df-0g 17342  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-grp 18846  df-minusg 18847

This theorem is referenced by:  grpissubg  19056