Theorem grpissubg 13799

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 2231	. . . . . 6 ⊢ (0g‘𝐻) = (0g‘𝐻)
5	3, 4	grpidcl 13630	. . . . 5 ⊢ (𝐻 ∈ Grp → (0g‘𝐻) ∈ 𝑆)
6		elex2 2819	. . . . 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 13608	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
10		mndmgm 13523	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
11	9, 10	syl 14	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mgm)
12		grpmnd 13608	. . . . . . . . . . 11 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
13		mndmgm 13523	. . . . . . . . . . 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 13462	. . . . . . 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 1273	. . . . . 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 2605	. . . . 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 529	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝐻 ∈ Grp)
29	22	sseq2i 3254	. . . . . . . . . . 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 6162	. . . . . . . . . . 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 6024	. . . . . . . . . . . . 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 2237	. . . . . . . . . . 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 2266	. . . . . . . . 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 2614	. . . . . . . 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 13678	. . . . . . 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 2231	. . . . . . . 8 ⊢ (invg‘𝐻) = (invg‘𝐻)
44	3, 43	grpinvcl 13649	. . . . . . 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 2309	. . . . 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 2605	. . 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 2231	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
50		eqid 2231	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
51	22, 49, 50	issubg2m 13794	. . . 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 1206	. 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 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200   × cxp 4723   ↾ cres 4727  ‘cfv 5326  (class class class)co 6018  Basecbs 13100  +_gcplusg 13178  0_gc0g 13357  Mgmcmgm 13455  Mndcmnd 13517  Grpcgrp 13601  inv_gcminusg 13602  SubGrpcsubg 13772

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-subg 13775

This theorem is referenced by:  resgrpisgrp  13800