Theorem mgmsscl 18578

Description: If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. Formerly part of proof of grpissubg 19084. (Contributed by AV, 17-Feb-2024.)

Hypotheses

Ref	Expression
mgmsscl.b	⊢ 𝐵 = (Base‘𝐺)
mgmsscl.s	⊢ 𝑆 = (Base‘𝐻)

Assertion

Ref	Expression
mgmsscl	⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)

Proof of Theorem mgmsscl

Step	Hyp	Ref	Expression
1		ovres 7562	. . 3 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐺)𝑌))
2	1	3ad2ant3 1135	. 2 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐺)𝑌))
3		simp1r 1199	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → 𝐻 ∈ Mgm)
4		simp3 1138	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆))
5		3anass 1094	. . . . 5 ⊢ ((𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ↔ (𝐻 ∈ Mgm ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)))
6	3, 4, 5	sylanbrc 583	. . . 4 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆))
7		mgmsscl.s	. . . . 5 ⊢ 𝑆 = (Base‘𝐻)
8		eqid 2730	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
9	7, 8	mgmcl 18576	. . . 4 ⊢ ((𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋(+g‘𝐻)𝑌) ∈ 𝑆)
10	6, 9	syl 17	. . 3 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐻)𝑌) ∈ 𝑆)
11		oveq 7400	. . . . . . 7 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐻)𝑌))
12	11	eleq1d 2814	. . . . . 6 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
13	12	eqcoms 2738	. . . . 5 ⊢ ((+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
14	13	adantl 481	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
15	14	3ad2ant2 1134	. . 3 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
16	10, 15	mpbird 257	. 2 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆)
17	2, 16	eqeltrrd 2830	1 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3922   × cxp 5644   ↾ cres 5648  ‘cfv 6519  (class class class)co 7394  Basecbs 17185  +_gcplusg 17226  Mgmcmgm 18571

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-xp 5652  df-res 5658  df-iota 6472  df-fv 6527  df-ov 7397  df-mgm 18573

This theorem is referenced by:  mndissubm  18740  grpissubg  19084