Theorem grpissubg 13402

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 (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.)

Hypotheses

Ref	Expression
grpissubg.b	⊢ 𝐵 = (Base‘𝐺)
grpissubg.s	⊢ 𝑆 = (Base‘𝐻)

Assertion

Ref	Expression
grpissubg	⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺)))

Proof of Theorem grpissubg

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ⊆ 𝐵)
2	1	adantl 277	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ⊆ 𝐵)
3		grpissubg.s	. . . . . 6 ⊢ 𝑆 = (Base‘𝐻)
4		eqid 2196	. . . . . 6 ⊢ (0g‘𝐻) = (0g‘𝐻)
5	3, 4	grpidcl 13233	. . . . 5 ⊢ (𝐻 ∈ Grp → (0g‘𝐻) ∈ 𝑆)
6		elex2 2779	. . . . 5 ⊢ ((0g‘𝐻) ∈ 𝑆 → ∃𝑤 𝑤 ∈ 𝑆)
7	5, 6	syl 14	. . . 4 ⊢ (𝐻 ∈ Grp → ∃𝑤 𝑤 ∈ 𝑆)
8	7	ad2antlr 489	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → ∃𝑤 𝑤 ∈ 𝑆)
9		grpmnd 13211	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
10		mndmgm 13126	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
11	9, 10	syl 14	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mgm)
12		grpmnd 13211	. . . . . . . . . . 11 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
13		mndmgm 13126	. . . . . . . . . . 11 ⊢ (𝐻 ∈ Mnd → 𝐻 ∈ Mgm)
14	12, 13	syl 14	. . . . . . . . . 10 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mgm)
15	11, 14	anim12i 338	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
16	15	adantr 276	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
17	16	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
18		simpr 110	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))))
19	18	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))))
20		simpr 110	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
21	20	anim1i 340	. . . . . . 7 ⊢ (((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆))
22		grpissubg.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
23	22, 3	mgmsscl 13065	. . . . . . 7 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)
24	17, 19, 21, 23	syl3anc 1249	. . . . . 6 ⊢ (((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)
25	24	ralrimiva 2570	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) → ∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)
26		simpl 109	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → 𝐺 ∈ Grp)
27	26	adantr 276	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝐺 ∈ Grp)
28		simplr 528	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝐻 ∈ Grp)
29	22	sseq2i 3211	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝐺))
30	29	biimpi 120	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ⊆ (Base‘𝐺))
31	30	adantr 276	. . . . . . . . 9 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ⊆ (Base‘𝐺))
32	31	adantl 277	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ⊆ (Base‘𝐺))
33		ovres 6067	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑦) = (𝑥(+g‘𝐺)𝑦))
34	33	adantl 277	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑦) = (𝑥(+g‘𝐺)𝑦))
35		oveq 5931	. . . . . . . . . . . . 13 ⊢ ((+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)) → (𝑥(+g‘𝐻)𝑦) = (𝑥((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑦))
36	35	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝑥(+g‘𝐻)𝑦) = (𝑥((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑦))
37	36	eqcomd 2202	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝑥((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑦) = (𝑥(+g‘𝐻)𝑦))
38	37	ad2antlr 489	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑦) = (𝑥(+g‘𝐻)𝑦))
39	34, 38	eqtr3d 2231	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
40	39	ralrimivva 2579	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
41	27, 28, 3, 32, 40	grpinvssd 13281	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → (𝑎 ∈ 𝑆 → ((invg‘𝐻)‘𝑎) = ((invg‘𝐺)‘𝑎)))
42	41	imp 124	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) → ((invg‘𝐻)‘𝑎) = ((invg‘𝐺)‘𝑎))
43		eqid 2196	. . . . . . . 8 ⊢ (invg‘𝐻) = (invg‘𝐻)
44	3, 43	grpinvcl 13252	. . . . . . 7 ⊢ ((𝐻 ∈ Grp ∧ 𝑎 ∈ 𝑆) → ((invg‘𝐻)‘𝑎) ∈ 𝑆)
45	44	ad4ant24 516	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) → ((invg‘𝐻)‘𝑎) ∈ 𝑆)
46	42, 45	eqeltrrd 2274	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) → ((invg‘𝐺)‘𝑎) ∈ 𝑆)
47	25, 46	jca 306	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ 𝑎 ∈ 𝑆) → (∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆 ∧ ((invg‘𝐺)‘𝑎) ∈ 𝑆))
48	47	ralrimiva 2570	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → ∀𝑎 ∈ 𝑆 (∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆 ∧ ((invg‘𝐺)‘𝑎) ∈ 𝑆))
49		eqid 2196	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
50		eqid 2196	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
51	22, 49, 50	issubg2m 13397	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑤 𝑤 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 (∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆 ∧ ((invg‘𝐺)‘𝑎) ∈ 𝑆))))
52	51	ad2antrr 488	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑤 𝑤 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 (∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆 ∧ ((invg‘𝐺)‘𝑎) ∈ 𝑆))))
53	2, 8, 48, 52	mpbir3and 1182	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ∈ (SubGrp‘𝐺))
54	53	ex 115	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157   × cxp 4662   ↾ cres 4666  ‘cfv 5259  (class class class)co 5925  Basecbs 12705  +_gcplusg 12782  0_gc0g 12960  Mgmcmgm 13058  Mndcmnd 13120  Grpcgrp 13204  inv_gcminusg 13205  SubGrpcsubg 13375

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-plusg 12795  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-minusg 13208  df-subg 13378

This theorem is referenced by:  resgrpisgrp  13403