Theorem grpinvssd 19057

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 2740	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
4	2, 3	grpinvcl 19027	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ 𝐵)
5	1, 4	sylan 579	. . . . 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 7455	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦))
10		oveq1 7455	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦))
11	9, 10	eqeq12d 2756	. . . . . 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 7456	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋))
13		oveq2 7456	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
14	12, 13	eqeq12d 2756	. . . . . 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 3647	. . . . 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 2740	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
18		eqid 2740	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	2, 17, 18, 3	grplinv 19029	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))
20	1, 19	sylan 579	. . . 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 4008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑀))
24		eqid 2740	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
25		eqid 2740	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
26		eqid 2740	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
27		eqid 2740	. . . . . . 7 ⊢ (invg‘𝑀) = (invg‘𝑀)
28	24, 25, 26, 27	grplinv 19029	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
29	21, 23, 28	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
30	21, 1, 2, 22, 7	grpidssd 19056	. . . . . 6 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑀) = (0g‘𝑆))
32	29, 31	eqtr2d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑆) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
33	16, 20, 32	3eqtrd 2784	. . 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 4009	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ (Base‘𝑀))
37	24, 27	grpinvcl 19027	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
38	21, 23, 37	syl2an2r 684	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
39	24, 25	grprcan 19013	. . . 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 1372	. . 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 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973  inv_gcminusg 18974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404  df-ov 7451  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977

This theorem is referenced by:  grpissubg  19186