Theorem mgmsscl 18331

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 18775. (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 7438	. . 3 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐺)𝑌))
2	1	3ad2ant3 1134	. 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 1197	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → 𝐻 ∈ Mgm)
4		simp3 1137	. . . . 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 2738	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
9	7, 8	mgmcl 18329	. . . 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 7281	. . . . . . 7 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐻)𝑌))
12	11	eleq1d 2823	. . . . . 6 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
13	12	eqcoms 2746	. . . . 5 ⊢ ((+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
14	13	adantl 482	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
15	14	3ad2ant2 1133	. . 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 256	. 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 2840	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887   × cxp 5587   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Mgmcmgm 18324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-res 5601  df-iota 6391  df-fv 6441  df-ov 7278  df-mgm 18326

This theorem is referenced by:  mndissubm  18446  grpissubg  18775