Theorem issubmgm 18631

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 6835	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
2	1	pweqd 4572	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3		fveq2 6835	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
4	3	oveqd 7377	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥(+g‘𝑀)𝑦))
5	4	eleq1d 2822	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
6	5	2ralbidv 3201	. . . . 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 3418	. . . 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 18622	. . . 4 ⊢ SubMgm = (𝑚 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡})
9		fvex 6848	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
10	9	pwex 5326	. . . . 5 ⊢ 𝒫 (Base‘𝑀) ∈ V
11	10	rabex 5285	. . . 4 ⊢ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡} ∈ V
12	7, 8, 11	fvmpt 6942	. . 3 ⊢ (𝑀 ∈ Mgm → (SubMgm‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡})
13	12	eleq2d 2823	. 2 ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡}))
14	9	elpw2 5280	. . . 4 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
15	14	anbi1i 625	. . 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 2826	. . . . . 6 ⊢ (𝑡 = 𝑆 → ((𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
17	16	raleqbi1dv 3309	. . . . 5 ⊢ (𝑡 = 𝑆 → (∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
18	17	raleqbi1dv 3309	. . . 4 ⊢ (𝑡 = 𝑆 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
19	18	elrab 3647	. . 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 3964	. . . 4 ⊢ (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝑀))
22		issubmgm.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
23	22	oveqi 7373	. . . . . 6 ⊢ (𝑥 + 𝑦) = (𝑥(+g‘𝑀)𝑦)
24	23	eleq1i 2828	. . . . 5 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
25	24	2ralbii 3112	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
26	21, 25	anbi12i 629	. . 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 303	. 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 287	1 ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400   ⊆ wss 3902  𝒫 cpw 4555  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  +_gcplusg 17181  Mgmcmgm 18567  SubMgmcsubmgm 18620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-submgm 18622

This theorem is referenced by:  issubmgm2  18632  rabsubmgmd  18633  submgmcl  18636  mgmhmima  18644  mgmhmeql  18645  submgmacs  18646