Theorem grpinvssd 18932

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 2733	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
4	2, 3	grpinvcl 18902	. . . . . 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 7359	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦))
10		oveq1 7359	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦))
11	9, 10	eqeq12d 2749	. . . . . 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 7360	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋))
13		oveq2 7360	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
14	12, 13	eqeq12d 2749	. . . . . 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 3585	. . . . 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 2733	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
18		eqid 2733	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	2, 17, 18, 3	grplinv 18904	. . . . 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 3930	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑀))
24		eqid 2733	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
25		eqid 2733	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
26		eqid 2733	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
27		eqid 2733	. . . . . . 7 ⊢ (invg‘𝑀) = (invg‘𝑀)
28	24, 25, 26, 27	grplinv 18904	. . . . . 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 18931	. . . . . 6 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑀) = (0g‘𝑆))
32	29, 31	eqtr2d 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑆) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
33	16, 20, 32	3eqtrd 2772	. . 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 3931	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ (Base‘𝑀))
37	24, 27	grpinvcl 18902	. . . . 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 18888	. . . 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 2113  ∀wral 3048   ⊆ wss 3898  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  +_gcplusg 17163  0_gc0g 17345  Grpcgrp 18848  inv_gcminusg 18849

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-riota 7309  df-ov 7355  df-0g 17347  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-grp 18851  df-minusg 18852

This theorem is referenced by:  grpissubg  19061