Theorem grpinvssd 17315

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 2610	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
4	2, 3	grpinvcl 17290	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ 𝐵)
5	1, 4	sylan 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ 𝐵)
6		simpr 476	. . . . 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 6556	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦))
10		oveq1 6556	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦))
11	9, 10	eqeq12d 2625	. . . . . 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 6557	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋))
13		oveq2 6557	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
14	12, 13	eqeq12d 2625	. . . . . 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 3294	. . . . 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 1317	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
17		eqid 2610	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
18		eqid 2610	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	2, 17, 18, 3	grplinv 17291	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))
20	1, 19	sylan 487	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))
21		grpidssd.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Grp)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ Grp)
23		grpidssd.c	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀))
24	23	sselda 3568	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑀))
25		eqid 2610	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
26		eqid 2610	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
27		eqid 2610	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
28		eqid 2610	. . . . . . 7 ⊢ (invg‘𝑀) = (invg‘𝑀)
29	25, 26, 27, 28	grplinv 17291	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
30	22, 24, 29	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
31	21, 1, 2, 23, 7	grpidssd 17314	. . . . . 6 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑀) = (0g‘𝑆))
33	30, 32	eqtr2d 2645	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑆) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
34	16, 20, 33	3eqtrd 2648	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
35	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐵 ⊆ (Base‘𝑀))
36	35, 5	sseldd 3569	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ (Base‘𝑀))
37	25, 28	grpinvcl 17290	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
38	22, 24, 37	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
39	25, 26	grprcan 17278	. . . 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	22, 36, 38, 24, 39	syl13anc 1320	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) ↔ ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋)))
41	34, 40	mpbid 221	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋))
42	41	ex 449	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Grpcgrp 17245  inv_gcminusg 17246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249

This theorem is referenced by:  grpissubg  17437