issubgrpd2 - Metamath Proof Explorer

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Theorem issubgrpd2 19160

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 4289	. 2 ⊢ (𝜑 → 𝐷 ≠ ∅)
4		issubgrpd.p	. . . . . . . 8 ⊢ (𝜑 → + = (+_g‘𝐼))
5	4	oveqd 7402	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+_g‘𝐼)𝑦))
6	5	ad2antrr 734	. . . . . 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 1127	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)
9	6, 8	eqeltrrd 2857	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑦 ∈ 𝐷) → (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷)
10	9	ralrimiva 3148	. . . 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 518	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∀𝑦 ∈ 𝐷 (𝑥(+_g‘𝐼)𝑦) ∈ 𝐷 ∧ ((inv_g‘𝐼)‘𝑥) ∈ 𝐷))
13	12	ralrimiva 3148	. 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 2756	. . . 4 ⊢ (Base‘𝐼) = (Base‘𝐼)
16		eqid 2756	. . . 4 ⊢ (+_g‘𝐼) = (+_g‘𝐼)
17		eqid 2756	. . . 4 ⊢ (inv_g‘𝐼) = (inv_g‘𝐼)
18	15, 16, 17	issubg2 19159	. . 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 1352	1 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1095 = wceq 1554 ∈ wcel 2136 ≠ wne 2951 ∀wral 3070 ⊆ wss 3899 ∅c0 4280 ‘cfv 6510 (class class class)co 7385 Basecbs 17221 ↾_s cress 17242 +_gcplusg 17262 0_gc0g 17444 Grpcgrp 18951 inv_gcminusg 18952 SubGrpcsubg 19138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-om 7836 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-er 8666 df-en 8917 df-dom 8918 df-sdom 8919 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-nn 12201 df-2 12270 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-0g 17446 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-grp 18954 df-minusg 18955 df-subg 19141

This theorem is referenced by: issubgrpd 19161 symgsssg 19483 symgfisg 19484 issubrgd 21229 dsmmsubg 21768 nsgmgclem 33551

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