Theorem issubmgm 18627

Description: Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)

Hypotheses

Ref	Expression
issubmgm.b	⊢ 𝐵 = (Base‘𝑀)
issubmgm.p	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
issubmgm	⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   + (𝑥,𝑦)

Proof of Theorem issubmgm

Dummy variables 𝑚 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6890	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
2	1	pweqd 4618	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3		fveq2 6890	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
4	3	oveqd 7428	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥(+g‘𝑀)𝑦))
5	4	eleq1d 2816	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
6	5	2ralbidv 3216	. . . . 5 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
7	2, 6	rabeqbidv 3447	. . . 4 ⊢ (𝑚 = 𝑀 → {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡} = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡})
8		df-submgm 18618	. . . 4 ⊢ SubMgm = (𝑚 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡})
9		fvex 6903	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
10	9	pwex 5377	. . . . 5 ⊢ 𝒫 (Base‘𝑀) ∈ V
11	10	rabex 5331	. . . 4 ⊢ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡} ∈ V
12	7, 8, 11	fvmpt 6997	. . 3 ⊢ (𝑀 ∈ Mgm → (SubMgm‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡})
13	12	eleq2d 2817	. 2 ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡}))
14	9	elpw2 5344	. . . 4 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
15	14	anbi1i 622	. . 3 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
16		eleq2 2820	. . . . . 6 ⊢ (𝑡 = 𝑆 → ((𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
17	16	raleqbi1dv 3331	. . . . 5 ⊢ (𝑡 = 𝑆 → (∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
18	17	raleqbi1dv 3331	. . . 4 ⊢ (𝑡 = 𝑆 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
19	18	elrab 3682	. . 3 ⊢ (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡} ↔ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
20		issubmgm.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
21	20	sseq2i 4010	. . . 4 ⊢ (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝑀))
22		issubmgm.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
23	22	oveqi 7424	. . . . . 6 ⊢ (𝑥 + 𝑦) = (𝑥(+g‘𝑀)𝑦)
24	23	eleq1i 2822	. . . . 5 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
25	24	2ralbii 3126	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
26	21, 25	anbi12i 625	. . 3 ⊢ ((𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
27	15, 19, 26	3bitr4i 302	. 2 ⊢ (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡} ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
28	13, 27	bitrdi 286	1 ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  {crab 3430   ⊆ wss 3947  𝒫 cpw 4601  ‘cfv 6542  (class class class)co 7411  Basecbs 17148  +_gcplusg 17201  Mgmcmgm 18563  SubMgmcsubmgm 18616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-submgm 18618

This theorem is referenced by:  issubmgm2  18628  rabsubmgmd  18629  submgmcl  18632  mgmhmima  18640  mgmhmeql  18641  submgmacs  18642