Theorem grpinvssd 13631

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 2229	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
4	2, 3	grpinvcl 13602	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ 𝐵)
5	1, 4	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ 𝐵)
6		simpr 110	. . . . 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 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦))
9		oveq1 6017	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦))
10		oveq1 6017	. . . . . . 7 ⊢ (𝑥 = ((invg‘𝑆)‘𝑋) → (𝑥(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦))
11	9, 10	eqeq12d 2244	. . . . . 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 6018	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋))
13		oveq2 6018	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑦) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
14	12, 13	eqeq12d 2244	. . . . . 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 2921	. . . . 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 1270	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋))
17		eqid 2229	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
18		eqid 2229	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
19	2, 17, 18, 3	grplinv 13604	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑆)𝑋) = (0g‘𝑆))
20	1, 19	sylan 283	. . . 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 3224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑀))
24		eqid 2229	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
25		eqid 2229	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
26		eqid 2229	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
27		eqid 2229	. . . . . . 7 ⊢ (invg‘𝑀) = (invg‘𝑀)
28	24, 25, 26, 27	grplinv 13604	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
29	21, 23, 28	syl2an2r 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋) = (0g‘𝑀))
30	21, 1, 2, 22, 7	grpidssd 13630	. . . . . 6 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))
31	30	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑀) = (0g‘𝑆))
32	29, 31	eqtr2d 2263	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (0g‘𝑆) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
33	16, 20, 32	3eqtrd 2266	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝑆)‘𝑋)(+g‘𝑀)𝑋) = (((invg‘𝑀)‘𝑋)(+g‘𝑀)𝑋))
34	21	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ Grp)
35	22	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐵 ⊆ (Base‘𝑀))
36	35, 5	sseldd 3225	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) ∈ (Base‘𝑀))
37	24, 27	grpinvcl 13602	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
38	21, 23, 37	syl2an2r 597	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑀)‘𝑋) ∈ (Base‘𝑀))
39	24, 25	grprcan 13591	. . . 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 1273	. . 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 147	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋))
42	41	ex 115	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → ((invg‘𝑆)‘𝑋) = ((invg‘𝑀)‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  ‘cfv 5321  (class class class)co 6010  Basecbs 13053  +_gcplusg 13131  0_gc0g 13310  Grpcgrp 13554  inv_gcminusg 13555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-cnex 8106  ax-resscn 8107  ax-1re 8109  ax-addrcl 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-inn 9127  df-2 9185  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-0g 13312  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558

This theorem is referenced by:  grpissubg  13752