issubgrpd2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > issubgrpd2		Structured version Visualization version GIF version

Theorem issubgrpd2 19050

Description: Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)

**Hypotheses**
Ref	Expression
issubgrpd.s	⊢ (𝜑 → 𝑆 = (𝐼 ↾_s 𝐷))
issubgrpd.z	⊢ (𝜑 → 0 = (0_g‘𝐼))
issubgrpd.p	⊢ (𝜑 → + = (+_g‘𝐼))
issubgrpd.ss	⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼))
issubgrpd.zcl	⊢ (𝜑 → 0 ∈ 𝐷)
issubgrpd.acl	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)
issubgrpd.ncl	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_g‘𝐼)‘𝑥) ∈ 𝐷)
issubgrpd.g	⊢ (𝜑 → 𝐼 ∈ Grp)

**Assertion**
Ref	Expression
issubgrpd2	⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼))

Distinct variable groups: 𝑥,𝑦, 0 𝑥,𝐷,𝑦 𝑥,𝐼,𝑦 𝑥, + ,𝑦 𝜑,𝑥,𝑦 𝑥,𝑆,𝑦

**Proof of Theorem issubgrpd2**
Step	Hyp	Ref	Expression
1		issubgrpd.ss	. 2 ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼))
2		issubgrpd.zcl	. . 3 ⊢ (𝜑 → 0 ∈ 𝐷)
3	2	ne0d 4287	. 2 ⊢ (𝜑 → 𝐷 ≠ ∅)
4		issubgrpd.p	. . . . . . . 8 ⊢ (𝜑 → + = (+_g‘𝐼))
5	4	oveqd 7358	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+_g‘𝐼)𝑦))
6	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) = (𝑥(+_g‘𝐼)𝑦))
7		issubgrpd.acl	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)
8	7	3expa 1118	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)
9	6, 8	eqeltrrd 2832	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷)
10	9	ralrimiva 3124	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∀𝑦 ∈ 𝐷 (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷)
11		issubgrpd.ncl	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((inv_g‘𝐼)‘𝑥) ∈ 𝐷)
12	10, 11	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∀𝑦 ∈ 𝐷 (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷 ∧ ((inv_g‘𝐼)‘𝑥) ∈ 𝐷))
13	12	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (∀𝑦 ∈ 𝐷 (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷 ∧ ((inv_g‘𝐼)‘𝑥) ∈ 𝐷))
14		issubgrpd.g	. . 3 ⊢ (𝜑 → 𝐼 ∈ Grp)
15		eqid 2731	. . . 4 ⊢ (Base‘𝐼) = (Base‘𝐼)
16		eqid 2731	. . . 4 ⊢ (+_g‘𝐼) = (+_g‘𝐼)
17		eqid 2731	. . . 4 ⊢ (inv_g‘𝐼) = (inv_g‘𝐼)
18	15, 16, 17	issubg2 19049	. . 3 ⊢ (𝐼 ∈ Grp → (𝐷 ∈ (SubGrp‘𝐼) ↔ (𝐷 ⊆ (Base‘𝐼) ∧ 𝐷 ≠ ∅ ∧ ∀𝑥 ∈ 𝐷 (∀𝑦 ∈ 𝐷 (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷 ∧ ((inv_g‘𝐼)‘𝑥) ∈ 𝐷))))
19	14, 18	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ (SubGrp‘𝐼) ↔ (𝐷 ⊆ (Base‘𝐼) ∧ 𝐷 ≠ ∅ ∧ ∀𝑥 ∈ 𝐷 (∀𝑦 ∈ 𝐷 (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷 ∧ ((inv_g‘𝐼)‘𝑥) ∈ 𝐷))))
20	1, 3, 13, 19	mpbir3and 1343	1 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ⊆ wss 3897 ∅c0 4278 ‘cfv 6476 (class class class)co 7341 Basecbs 17115 ↾_s cress 17136 +_gcplusg 17156 0_gc0g 17338 Grpcgrp 18841 inv_gcminusg 18842 SubGrpcsubg 19028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-0g 17340 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-grp 18844 df-minusg 18845 df-subg 19031

This theorem is referenced by: issubgrpd 19051 symgsssg 19374 symgfisg 19375 issubrgd 21118 dsmmsubg 21675 nsgmgclem 33368

W3C validator