Theorem mgmsscl 18677

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 19183. (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 7603	. . 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 1198	. . . . 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 2736	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
9	7, 8	mgmcl 18675	. . . 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 7441	. . . . . . 7 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐻)𝑌))
12	11	eleq1d 2825	. . . . . 6 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
13	12	eqcoms 2744	. . . . 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 2841	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 1538   ∈ wcel 2107   ⊆ wss 3964   × cxp 5688   ↾ cres 5692  ‘cfv 6566  (class class class)co 7435  Basecbs 17251  +_gcplusg 17304  Mgmcmgm 18670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-sbc 3793  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-xp 5696  df-res 5702  df-iota 6519  df-fv 6574  df-ov 7438  df-mgm 18672

This theorem is referenced by:  mndissubm  18839  grpissubg  19183